# Coverage for pygeodesy/ellipsoids.py : 98%

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2222 2223 2224 2225 2226 2227 2228 2229 2230 2231 2232 2233 2234 2235 2236 2237 2238 2239 2240 2241 2242 2243 2244 2245 2246 2247 2248 2249 2250 2251 2252 2253 2254 2255 2256 2257 2258 2259 2260 2261 2262 2263 2264 2265 2266 2267 2268 2269 2270 2271 2272 2273 2274 2275 2276 2277 2278 2279 2280 2281 2282 2283 2284 2285 2286 2287 2288 2289 2290 2291 2292 2293 2294 2295 2296 2297 2298 2299 2300 2301 2302 2303 2304 2305 2306 2307 2308 2309 2310 2311 2312 2313 2314 2315 | # -*- coding: utf-8 -*- u'''Ellipsoidal and spherical earth models. Classes L{a_f2Tuple}, L{Ellipsoid} and L{Ellipsoid2}, an L{Ellipsoids} registry and 2 dozen functions to convert I{equatorial} radius, I{polar} radius, I{eccentricities}, I{flattenings} and I{inverse flattening}. See module L{datums} for L{Datum} and L{Transform} information and other details. Following is the list of predefined L{Ellipsoid}s, all instantiated lazily. @var Ellipsoids.Airy1830: Ellipsoid(name='Airy1830', a=6377563.396, b=6356256.90923729, f_=299.3249646, f=0.00334085, f2=0.00335205, n=0.00167322, e=0.08167337, e2=0.00667054, e22=0.00671533, e32=0.00334643, A=6366914.60892522, L=10001126.0807165, R1=6370461.23374576, R2=6370459.65470808, R3=6370453.30994572, Rbiaxial=6366919.065224, Rtriaxial=6372243.45317691) @var Ellipsoids.AiryModified: Ellipsoid(name='AiryModified', a=6377340.189, b=6356034.44793853, f_=299.3249646, f=0.00334085, f2=0.00335205, n=0.00167322, e=0.08167337, e2=0.00667054, e22=0.00671533, e32=0.00334643, A=6366691.77461988, L=10000776.05340819, R1=6370238.27531284, R2=6370236.69633043, R3=6370230.35179013, Rbiaxial=6366696.2307627, Rtriaxial=6372020.43236847) @var Ellipsoids.ATS1977: Ellipsoid(name='ATS1977', a=6378135, b=6356750.30492159, f_=298.257, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181922, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367447.14116695, L=10001962.58040571, R1=6371006.7683072, R2=6371005.17780873, R3=6370998.78689182, Rbiaxial=6367451.62986519, Rtriaxial=6372795.55363648) @var Ellipsoids.Australia1966: Ellipsoid(name='Australia1966', a=6378160, b=6356774.71919531, f_=298.25, f=0.00335289, f2=0.00336417, n=0.00167926, e=0.08182018, e2=0.00669454, e22=0.00673966, e32=0.00335851, A=6367471.84853228, L=10002001.39064442, R1=6371031.5730651, R2=6371029.9824858, R3=6371023.59124344, Rbiaxial=6367476.337459, Rtriaxial=6372820.40754721) @var Ellipsoids.Bessel1841: Ellipsoid(name='Bessel1841', a=6377397.155, b=6356078.962818, f_=299.1528128, f=0.00334277, f2=0.00335398, n=0.00167418, e=0.08169683, e2=0.00667437, e22=0.00671922, e32=0.00334836, A=6366742.52023395, L=10000855.76443237, R1=6370291.09093933, R2=6370289.51012659, R3=6370283.15821523, Rbiaxial=6366746.98155108, Rtriaxial=6372074.29334012) @var Ellipsoids.BesselModified: Ellipsoid(name='BesselModified', a=6377492.018, b=6356173.5087127, f_=299.1528128, f=0.00334277, f2=0.00335398, n=0.00167418, e=0.08169683, e2=0.00667437, e22=0.00671922, e32=0.00334836, A=6366837.22474766, L=10001004.52593463, R1=6370385.84823756, R2=6370384.26740131, R3=6370377.91539546, Rbiaxial=6366841.68613115, Rtriaxial=6372169.07716325) @var Ellipsoids.CGCS2000: Ellipsoid(name='CGCS2000', a=6378137, b=6356752.31414036, f_=298.2572221, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367449.14577105, L=10001965.72923046, R1=6371008.77138012, R2=6371007.18088352, R3=6371000.78997414, Rbiaxial=6367453.63446401, Rtriaxial=6372797.55593326) @var Ellipsoids.Clarke1866: Ellipsoid(name='Clarke1866', a=6378206.4, b=6356583.8, f_=294.97869821, f=0.00339008, f2=0.00340161, n=0.00169792, e=0.08227185, e2=0.00676866, e22=0.00681478, e32=0.00339582, A=6367399.68916978, L=10001888.04298286, R1=6370998.86666667, R2=6370997.240633, R3=6370990.70659881, Rbiaxial=6367404.2783313, Rtriaxial=6372807.62791066) @var Ellipsoids.Clarke1880: Ellipsoid(name='Clarke1880', a=6378249.145, b=6356514.86954978, f_=293.465, f=0.00340756, f2=0.00341921, n=0.00170669, e=0.0824834, e2=0.00680351, e22=0.00685012, e32=0.00341337, A=6367386.64398051, L=10001867.55164747, R1=6371004.38651659, R2=6371002.74366963, R3=6370996.1419165, Rbiaxial=6367391.2806777, Rtriaxial=6372822.52526083) @var Ellipsoids.Clarke1880IGN: Ellipsoid(name='Clarke1880IGN', a=6378249.2, b=6356515, f_=293.46602129, f=0.00340755, f2=0.0034192, n=0.00170668, e=0.08248326, e2=0.00680349, e22=0.00685009, e32=0.00341336, A=6367386.73667336, L=10001867.69724907, R1=6371004.46666667, R2=6371002.82383112, R3=6370996.22212395, Rbiaxial=6367391.37333829, Rtriaxial=6372822.59907505) @var Ellipsoids.Clarke1880Mod: Ellipsoid(name='Clarke1880Mod', a=6378249.145, b=6356514.96639549, f_=293.46630766, f=0.00340755, f2=0.0034192, n=0.00170668, e=0.08248322, e2=0.00680348, e22=0.00685009, e32=0.00341335, A=6367386.69236201, L=10001867.62764495, R1=6371004.4187985, R2=6371002.77596616, R3=6370996.17427195, Rbiaxial=6367391.32901784, Rtriaxial=6372822.5494103) @var Ellipsoids.CPM1799: Ellipsoid(name='CPM1799', a=6375738.7, b=6356671.92557493, f_=334.39, f=0.00299052, f2=0.00299949, n=0.0014975, e=0.07727934, e2=0.0059721, e22=0.00600798, e32=0.00299499, A=6366208.88184784, L=10000017.52721564, R1=6369383.10852498, R2=6369381.8434158, R3=6369376.76247022, Rbiaxial=6366212.45090321, Rtriaxial=6370977.3559758) @var Ellipsoids.Delambre1810: Ellipsoid(name='Delambre1810', a=6376428, b=6355957.92616372, f_=311.5, f=0.00321027, f2=0.00322061, n=0.00160772, e=0.08006397, e2=0.00641024, e22=0.0064516, e32=0.00321543, A=6366197.07684334, L=9999998.98395793, R1=6369604.64205457, R2=6369603.18419749, R3=6369597.32739068, Rbiaxial=6366201.19059818, Rtriaxial=6371316.64722284) @var Ellipsoids.Engelis1985: Ellipsoid(name='Engelis1985', a=6378136.05, b=6356751.32272154, f_=298.2566, f=0.00335282, f2=0.0033641, n=0.00167922, e=0.08181928, e2=0.00669439, e22=0.00673951, e32=0.00335844, A=6367448.17507971, L=10001964.20447208, R1=6371007.80757385, R2=6371006.21707085, R3=6370999.82613573, Rbiaxial=6367452.66379074, Rtriaxial=6372796.59560563) @var Ellipsoids.Everest1969: Ellipsoid(name='Everest1969', a=6377295.664, b=6356094.667915, f_=300.8017, f=0.00332445, f2=0.00333554, n=0.00166499, e=0.08147298, e2=0.00663785, e22=0.0066822, e32=0.00332998, A=6366699.57839501, L=10000788.3115495, R1=6370228.665305, R2=6370227.10178537, R3=6370220.81951618, Rbiaxial=6366703.99082487, Rtriaxial=6372002.02812501) @var Ellipsoids.Everest1975: Ellipsoid(name='Everest1975', a=6377299.151, b=6356098.14512013, f_=300.8017255, f=0.00332445, f2=0.00333554, n=0.00166499, e=0.08147298, e2=0.00663785, e22=0.0066822, e32=0.00332997, A=6366703.06049924, L=10000793.78122603, R1=6370232.14904004, R2=6370230.58551983, R3=6370224.30324826, Rbiaxial=6366707.47293076, Rtriaxial=6372005.51267879) @var Ellipsoids.Fisher1968: Ellipsoid(name='Fisher1968', a=6378150, b=6356768.33724438, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, A=6367463.65604381, L=10001988.52191361, R1=6371022.77908146, R2=6371021.18903735, R3=6371014.79995035, Rbiaxial=6367468.14345752, Rtriaxial=6372811.30979281) @var Ellipsoids.GEM10C: Ellipsoid(name='GEM10C', a=6378137, b=6356752.31424783, f_=298.2572236, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367449.14582474, L=10001965.7293148, R1=6371008.77141594, R2=6371007.18091936, R3=6371000.79001005, Rbiaxial=6367453.63451765, Rtriaxial=6372797.55596006) @var Ellipsoids.GPES: Ellipsoid(name='GPES', a=6378135, b=6378135, f_=0, f=0, f2=0, n=0, e=0, e2=0, e22=0, e32=0, A=6378135, L=10018751.02980197, R1=6378135, R2=6378135, R3=6378135, Rbiaxial=6378135, Rtriaxial=6378135) @var Ellipsoids.GRS67: Ellipsoid(name='GRS67', a=6378160, b=6356774.51609071, f_=298.24716743, f=0.00335292, f2=0.0033642, n=0.00167928, e=0.08182057, e2=0.00669461, e22=0.00673973, e32=0.00335854, A=6367471.74706533, L=10002001.2312605, R1=6371031.50536357, R2=6371029.91475409, R3=6371023.52339015, Rbiaxial=6367476.23607738, Rtriaxial=6372820.3568989) @var Ellipsoids.GRS80: Ellipsoid(name='GRS80', a=6378137, b=6356752.31414035, f_=298.2572221, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367449.14577104, L=10001965.72923046, R1=6371008.77138012, R2=6371007.18088351, R3=6371000.78997414, Rbiaxial=6367453.634464, Rtriaxial=6372797.55593326) @var Ellipsoids.Helmert1906: Ellipsoid(name='Helmert1906', a=6378200, b=6356818.16962789, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, A=6367513.57227074, L=10002066.93013953, R1=6371072.7232093, R2=6371071.13315272, R3=6371064.74401563, Rbiaxial=6367518.05971963, Rtriaxial=6372861.26794141) @var Ellipsoids.IAU76: Ellipsoid(name='IAU76', a=6378140, b=6356755.28815753, f_=298.257, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181922, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367452.13278844, L=10001970.42122641, R1=6371011.76271918, R2=6371010.17221946, R3=6371003.78129754, Rbiaxial=6367456.6214902, Rtriaxial=6372800.54945074) @var Ellipsoids.IERS1989: Ellipsoid(name='IERS1989', a=6378136, b=6356751.30156878, f_=298.257, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181922, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367448.13949125, L=10001964.14856985, R1=6371007.76718959, R2=6371006.17669088, R3=6370999.78577297, Rbiaxial=6367452.62819019, Rtriaxial=6372796.55279934) @var Ellipsoids.IERS1992TOPEX: Ellipsoid(name='IERS1992TOPEX', a=6378136.3, b=6356751.61659215, f_=298.25722356, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367448.44699641, L=10001964.63159782, R1=6371008.07219738, R2=6371006.48170097, R3=6371000.09079236, Rbiaxial=6367452.93568883, Rtriaxial=6372796.85654541) @var Ellipsoids.IERS2003: Ellipsoid(name='IERS2003', a=6378136.6, b=6356751.85797165, f_=298.25642, f=0.00335282, f2=0.0033641, n=0.00167922, e=0.0818193, e2=0.0066944, e22=0.00673951, e32=0.00335844, A=6367448.71771058, L=10001965.05683465, R1=6371008.35265722, R2=6371006.76215217, R3=6371000.37120877, Rbiaxial=6367453.20642742, Rtriaxial=6372797.14192686) @var Ellipsoids.Intl1924: Ellipsoid(name='Intl1924', a=6378388, b=6356911.94612795, f_=297, f=0.003367, f2=0.00337838, n=0.00168634, e=0.08199189, e2=0.00672267, e22=0.00676817, e32=0.00337267, A=6367654.50005758, L=10002288.29898944, R1=6371229.31537598, R2=6371227.71133444, R3=6371221.26587487, Rbiaxial=6367659.02704315, Rtriaxial=6373025.77129687) @var Ellipsoids.Intl1967: Ellipsoid(name='Intl1967', a=6378157.5, b=6356772.2, f_=298.24961539, f=0.0033529, f2=0.00336418, n=0.00167926, e=0.08182023, e2=0.00669455, e22=0.00673967, e32=0.00335852, A=6367469.33894446, L=10001997.44859308, R1=6371029.06666667, R2=6371027.47608389, R3=6371021.08482752, Rbiaxial=6367473.827881, Rtriaxial=6372817.9027631) @var Ellipsoids.Krassovski1940: Ellipsoid(name='Krassovski1940', a=6378245, b=6356863.01877305, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, A=6367558.49687498, L=10002137.49754285, R1=6371117.67292435, R2=6371116.08285656, R3=6371109.69367439, Rbiaxial=6367562.98435553, Rtriaxial=6372906.23027515) @var Ellipsoids.Krassowsky1940: Ellipsoid(name='Krassowsky1940', a=6378245, b=6356863.01877305, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, A=6367558.49687498, L=10002137.49754285, R1=6371117.67292435, R2=6371116.08285656, R3=6371109.69367439, Rbiaxial=6367562.98435553, Rtriaxial=6372906.23027515) @var Ellipsoids.Maupertuis1738: Ellipsoid(name='Maupertuis1738', a=6397300, b=6363806.28272251, f_=191, f=0.0052356, f2=0.00526316, n=0.00262467, e=0.10219488, e2=0.01044379, e22=0.01055402, e32=0.00524931, A=6380564.13011837, L=10022566.69846922, R1=6386135.42757417, R2=6386131.54144847, R3=6386115.8862823, Rbiaxial=6380575.11882818, Rtriaxial=6388943.03218495) @var Ellipsoids.Mercury1960: Ellipsoid(name='Mercury1960', a=6378166, b=6356784.28360711, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, A=6367479.62923643, L=10002013.61254591, R1=6371038.76120237, R2=6371037.17115427, R3=6371030.78205124, Rbiaxial=6367484.1166614, Rtriaxial=6372827.29640037) @var Ellipsoids.Mercury1968Mod: Ellipsoid(name='Mercury1968Mod', a=6378150, b=6356768.33724438, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, A=6367463.65604381, L=10001988.52191361, R1=6371022.77908146, R2=6371021.18903735, R3=6371014.79995035, Rbiaxial=6367468.14345752, Rtriaxial=6372811.30979281) @var Ellipsoids.NWL1965: Ellipsoid(name='NWL1965', a=6378145, b=6356759.76948868, f_=298.25, f=0.00335289, f2=0.00336417, n=0.00167926, e=0.08182018, e2=0.00669454, e22=0.00673966, e32=0.00335851, A=6367456.87366841, L=10001977.86818326, R1=6371016.58982956, R2=6371014.999254, R3=6371008.60802667, Rbiaxial=6367461.36258457, Rtriaxial=6372805.42010473) @var Ellipsoids.OSU86F: Ellipsoid(name='OSU86F', a=6378136.2, b=6356751.51693008, f_=298.2572236, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367448.3471653, L=10001964.47478349, R1=6371007.97231003, R2=6371006.38181364, R3=6370999.99090513, Rbiaxial=6367452.83585765, Rtriaxial=6372796.75662978) @var Ellipsoids.OSU91A: Ellipsoid(name='OSU91A', a=6378136.3, b=6356751.6165948, f_=298.2572236, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367448.44699773, L=10001964.63159991, R1=6371008.07219827, R2=6371006.48170186, R3=6371000.09079324, Rbiaxial=6367452.93569015, Rtriaxial=6372796.85654607) @var Ellipsoids.Plessis1817: Ellipsoid(name='Plessis1817', a=6376523, b=6355862.93325557, f_=308.64, f=0.00324002, f2=0.00325055, n=0.00162264, e=0.08043347, e2=0.00646954, e22=0.00651167, e32=0.00324527, A=6366197.15710739, L=9999999.11003639, R1=6369636.31108519, R2=6369634.82608583, R3=6369628.85999668, Rbiaxial=6366201.34758009, Rtriaxial=6371364.26393357) @var Ellipsoids.PZ90: Ellipsoid(name='PZ90', a=6378136, b=6356751.36174571, f_=298.2578393, f=0.0033528, f2=0.00336408, n=0.00167922, e=0.08181911, e2=0.00669437, e22=0.00673948, e32=0.00335842, A=6367448.16955443, L=10001964.19579298, R1=6371007.78724857, R2=6371006.1967588, R3=6370999.80587691, Rbiaxial=6367452.65822809, Rtriaxial=6372796.56780569) @var Ellipsoids.SGS85: Ellipsoid(name='SGS85', a=6378136, b=6356751.30156878, f_=298.257, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181922, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367448.13949125, L=10001964.14856985, R1=6371007.76718959, R2=6371006.17669087, R3=6370999.78577297, Rbiaxial=6367452.62819019, Rtriaxial=6372796.55279934) @var Ellipsoids.SoAmerican1969: Ellipsoid(name='SoAmerican1969', a=6378160, b=6356774.71919531, f_=298.25, f=0.00335289, f2=0.00336417, n=0.00167926, e=0.08182018, e2=0.00669454, e22=0.00673966, e32=0.00335851, A=6367471.84853228, L=10002001.39064442, R1=6371031.5730651, R2=6371029.98248581, R3=6371023.59124344, Rbiaxial=6367476.337459, Rtriaxial=6372820.40754721) @var Ellipsoids.Sphere: Ellipsoid(name='Sphere', a=6371008.771415, b=6371008.771415, f_=0, f=0, f2=0, n=0, e=0, e2=0, e22=0, e32=0, A=6371008.771415, L=10007557.17611675, R1=6371008.771415, R2=6371008.771415, R3=6371008.771415, Rbiaxial=6371008.771415, Rtriaxial=6371008.771415) @var Ellipsoids.SphereAuthalic: Ellipsoid(name='SphereAuthalic', a=6371000, b=6371000, f_=0, f=0, f2=0, n=0, e=0, e2=0, e22=0, e32=0, A=6371000, L=10007543.39801029, R1=6371000, R2=6371000, R3=6371000, Rbiaxial=6371000, Rtriaxial=6371000) @var Ellipsoids.SpherePopular: Ellipsoid(name='SpherePopular', a=6378137, b=6378137, f_=0, f=0, f2=0, n=0, e=0, e2=0, e22=0, e32=0, A=6378137, L=10018754.17139462, R1=6378137, R2=6378137, R3=6378137, Rbiaxial=6378137, Rtriaxial=6378137) @var Ellipsoids.Struve1860: Ellipsoid(name='Struve1860', a=6378298.3, b=6356657.14266956, f_=294.73, f=0.00339294, f2=0.00340449, n=0.00169935, e=0.0823065, e2=0.00677436, e22=0.00682056, e32=0.00339869, A=6367482.31832549, L=10002017.83655714, R1=6371084.58088985, R2=6371082.95208988, R3=6371076.40691418, Rbiaxial=6367486.91530791, Rtriaxial=6372894.90029454) @var Ellipsoids.WGS60: Ellipsoid(name='WGS60', a=6378165, b=6356783.28695944, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, A=6367478.63091189, L=10002012.0443814, R1=6371037.76231981, R2=6371036.17227197, R3=6371029.78316994, Rbiaxial=6367483.11833616, Rtriaxial=6372826.29723739) @var Ellipsoids.WGS66: Ellipsoid(name='WGS66', a=6378145, b=6356759.76948868, f_=298.25, f=0.00335289, f2=0.00336417, n=0.00167926, e=0.08182018, e2=0.00669454, e22=0.00673966, e32=0.00335851, A=6367456.87366841, L=10001977.86818326, R1=6371016.58982956, R2=6371014.999254, R3=6371008.60802667, Rbiaxial=6367461.36258457, Rtriaxial=6372805.42010473) @var Ellipsoids.WGS72: Ellipsoid(name='WGS72', a=6378135, b=6356750.52001609, f_=298.26, f=0.00335278, f2=0.00336406, n=0.0016792, e=0.08181881, e2=0.00669432, e22=0.00673943, e32=0.0033584, A=6367447.24862383, L=10001962.74919858, R1=6371006.84000536, R2=6371005.24953886, R3=6370998.8587507, Rbiaxial=6367451.7372317, Rtriaxial=6372795.60727472) @var Ellipsoids.WGS84: Ellipsoid(name='WGS84', a=6378137, b=6356752.31424518, f_=298.25722356, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367449.14582341, L=10001965.72931272, R1=6371008.77141506, R2=6371007.18091847, R3=6371000.79000916, Rbiaxial=6367453.63451633, Rtriaxial=6372797.5559594) ''' # make sure int/int division yields float quotient, see .basics from __future__ import division as _; del _ # PYCHOK semicolon from pygeodesy.basics import copysign0, isint from pygeodesy.constants import EPS, EPS0, EPS02, EPS1, INF, NINF, PI4, PI_2, PI_3, R_M, R_MA, R_FM, \ _EPSqrt, _EPStol as _TOL, _floatuple as _T, _isfinite, _SQRT2_2, \ _0_0s, _0_0, _0_5, _1_0, _1_EPS, _2_0, _4_0, _90_0, \ _0_25, _3_0 # PYCHOK used! from pygeodesy.errors import _AssertionError, _ValueError, _xkwds_not from pygeodesy.fmath import cbrt, cbrt2, fdot, Fhorner, fpowers, Fsum, hypot, hypot_, \ hypot1, hypot2, sqrt3 # from pygeodesy.fsums import Fsum # from .fmath from pygeodesy.interns import NN, _a_, _Airy1830_, _AiryModified_, _b_, _Bessel1841_, \ _Clarke1866_, _Clarke1880IGN_, _DOT_, _f_, _GRS80_, _height_, \ _Intl1924_, _incompatible_, _invalid_, _Krassovski1940_, \ _Krassowsky1940_, _meridional_, _lat_, _negative_, _not_finite_, \ _vs_, _prime_vertical_, _radius_, _Sphere_, _SPACE_, _WGS72_, _WGS84_ from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS from pygeodesy.named import _lazyNamedEnumItem as _lazy, _NamedEnum, \ _NamedEnumItem, _NamedTuple, _Pass from pygeodesy.namedTuples import Distance2Tuple, Vector3Tuple, Vector4Tuple from pygeodesy.props import deprecated_Property_RO, Property_RO, property_doc_, property_RO from pygeodesy.streprs import Fmt, fstr, instr, strs, unstr from pygeodesy.units import Bearing_, Distance, Float, Float_, Height, Lam_, Lat, Meter, \ Meter2, Meter3, Phi, Phi_, Radius, Radius_, Scalar from pygeodesy.utily import atand, atan2b, atan2d, degrees90, m2radians, radians2m, sincos2d from math import asinh, atan, atanh, cos, degrees, exp, radians, sin, sinh, sqrt, tan __all__ = _ALL_LAZY.ellipsoids __version__ = '22.11.02' _f_0_0 = Float(f =_0_0) # zero flattening _f__0_0 = Float(f_=_0_0) # zero inverse flattening # see U{WGS84_f<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Constants.html>} _f__WGS84 = Float(f_=_1_0 / (1000000000 / 298257223563)) # 298.25722356299997 vs 298.257223563 def _aux(lat, inverse, auxLat, clip=90): '''Return a named auxiliary latitude in C{degrees}. ''' return Lat(lat, clip=clip, name=_lat_ if inverse else auxLat.__name__) def _s2_c2(phi): '''(INTERNAL) Return 2-tuple C{(sin(B{phi})**2, cos(B{phi})**2)}. ''' if phi: s2 = sin(phi)**2 if s2 > 0: c2 = _1_0 - s2 if c2 > 0: if c2 < _1_0: return s2, c2 else: return _1_0, _0_0 # phi == PI_2 return _0_0, _1_0 # phi == 0 class a_f2Tuple(_NamedTuple): '''2-Tuple C{(a, f)} specifying an ellipsoid by I{equatorial} radius C{a} in C{meter} and scalar I{flattening} C{f}. @see: Class L{Ellipsoid2}. ''' _Names_ = (_a_, _f_) # name 'f' not 'f_' _Units_ = (_Pass, _Pass) def __new__(cls, a, f, **name): '''New L{a_f2Tuple} ellipsoid specification. @arg a: Equatorial radius (C{scalar} > 0). @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @kwarg name: Optional name (C{str}). @return: An L{a_f2Tuple}C{(a, f)} instance. @raise UnitError: Invalid B{C{a}} or B{C{f}}. @note: C{abs(B{f}) < EPS} is forced to C{B{f}=0}, I{spherical}. Negative C{B{f}} produces a I{prolate} ellipsoid. ''' a = Radius_(a=a) # low=EPS, high=None f = Float_( f=f, low=None, high=EPS1) if abs(f) < EPS: # force spherical f = _f_0_0 return _NamedTuple.__new__(cls, a, f, **name) @Property_RO def b(self): '''Get the I{polar} radius (C{meter}), M{a * (1 - f)}. ''' return a_f2b(self.a, self.f) # PYCHOK .a and .f def ellipsoid(self, name=NN): '''Return an L{Ellipsoid} for this 2-tuple C{(a, f)}. @raise NameError: A registered C{ellipsoid} with the same B{C{name}} already exists. ''' return Ellipsoid(self.a, f=self.f, name=name or self.name) # PYCHOK .a and .f @Property_RO def f_(self): '''Get the I{inverse} flattening (C{float}), M{1 / f} == M{a / (a - b)}. ''' return f2f_(self.f) # PYCHOK .f class Circle4Tuple(_NamedTuple): '''4-Tuple C{(radius, height, lat, beta)} of the C{radius} and C{height}, both conventionally in C{meter} of a parallel I{circle of latitude} at (geodetic) latitude C{lat} and the I{parametric (or reduced) auxiliary latitude} C{beta}, both in C{degrees90}. The C{height} is the (signed) distance along the z-axis between the parallel and the equator. At near-polar C{lat}s, the C{radius} is C{0}, the C{height} is the ellipsoid's (signed) polar radius and C{beta} equals C{lat}. ''' _Names_ = (_radius_, _height_, _lat_, 'beta') _Units_ = ( Radius, Height, Lat, Lat) class Curvature2Tuple(_NamedTuple): '''2-Tuple C{(meridional, prime_vertical)} of radii of curvature, both in C{meter}, conventionally. ''' _Names_ = (_meridional_, _prime_vertical_) _Units_ = ( Meter, Meter) @property_RO def transverse(self): '''Get this I{prime_vertical}, aka I{transverse} radius of curvature. ''' return self.prime_vertical class Ellipsoid(_NamedEnumItem): '''Ellipsoid with I{equatorial} and I{polar} radii, I{flattening}, I{inverse flattening} and other, often used, I{cached} attributes, supporting I{oblate} and I{prolate} ellipsoidal and I{spherical} earth models. ''' _a = 0 # equatorial radius, semi-axis (C{meter}) _b = 0 # polar radius, semi-axis (C{meter}): a * (f - 1) / f _f = 0 # (1st) flattening: (a - b) / a _f_ = 0 # inverse flattening: 1 / f = a / (a - b) _KsOrder = 8 # Krüger series order (4, 6 or 8) def __init__(self, a, b=None, f_=None, f=None, name=NN): '''New L{Ellipsoid} from the I{equatorial} radius and either the I{polar} radius or I{inverse flattening} or I{flattening}. @arg a: Equatorial radius, semi-axis (C{meter}). @arg b: Optional polar radius, semi-axis (C{meter}). @arg f_: Inverse flattening: M{a / (a - b)} (C{float} >>> 1.0). @arg f: Flattening: M{(a - b) / a} (C{float}, near zero for spherical). @kwarg name: Optional, unique name (C{str}). @raise NameError: Ellipsoid with the same B{C{name}} already exists. @raise ValueError: Invalid B{C{a}}, B{C{b}}, B{C{f_}} or B{C{f}} or B{C{f_}} and B{C{f}} are incompatible. @note: M{abs(f_) > 1 / EPS} or M{abs(1 / f_) < EPS} is forced to M{1 / f_ = 0}, spherical. ''' ff_ = f, f_ # assertion below try: a = Radius_(a=a) # low=EPS if not _isfinite(a): raise ValueError(_SPACE_(_a_, _not_finite_)) if b: # not in (_0_0, None) b = Radius_(b=b) # low=EPS f = a_b2f(a, b) if f is None else Float(f=f) f_ = f2f_(f) if f_ is None else Float(f_=f_) elif f is not None: f = Float(f=f) b = a_f2b(a, f) f_ = f2f_(f) if f_ is None else Float(f_=f_) elif f_: f_ = Float(f_=f_) b = a_f_2b(a, f_) # a * (f_ - 1) / f_ f = f_2f(f_) else: # only a, spherical f = f_ = 0 b = a # superfluous if not f < _1_0: # sanity check, see .ecef.Ecef.__init__ raise ValueError(_SPACE_(_f_, _invalid_)) if not _isfinite(b): raise ValueError(_SPACE_(_b_, _not_finite_)) if abs(f) < EPS or a == b or not f_: # spherical b = a f = _f_0_0 f_ = _f__0_0 except (TypeError, ValueError) as x: d = _xkwds_not(None, b=b, f_=f_, f=f) t = instr(self, a=a, name=name, **d) raise _ValueError(t, cause=x) self._a = a self._b = b self._f = f self._f_ = f_ self._register(Ellipsoids, name) if f and f_: # see .test/testEllipsoidal.py d = dict(eps=_TOL) if None in ff_: # both f_ and f given d.update(Error=_ValueError, txt=_incompatible_) self._assert(_1_0 / f, f_=f_, **d) self._assert(_1_0 / f_, f =f, **d) self._assert(self.b2_a2, e21=self.e21, eps=EPS) def __eq__(self, other): '''Compare this and an other ellipsoid. @arg other: The other ellipsoid (L{Ellipsoid} or L{Ellipsoid2}). @return: C{True} if equal, C{False} otherwise. ''' return self is other or (isinstance(other, Ellipsoid) and self.a == other.a and (self.f == other.f or self.b == other.b)) @Property_RO def a(self): '''Get the I{equatorial} radius, semi-axis (C{meter}). ''' return self._a equatoradius = a # = Requatorial @Property_RO def a2(self): '''Get the I{equatorial} radius I{squared} (C{meter} I{squared}), M{a**2}. ''' return Meter2(a2=self.a**2) @Property_RO def a2_(self): '''Get the inverse of the I{equatorial} radius I{squared} (C{meter} I{squared}), M{1 / a**2}. ''' return Float(a2_=_1_0 / self.a2) @Property_RO def a_b(self): '''Get the ratio I{equatorial} over I{polar} radius (C{float}), M{a / b} == M{1 / (1 - f)}. ''' return Float(a_b=self.a / self.b if self.f else _1_0) @Property_RO def a2_b(self): '''Get the I{polar} meridional (or polar) radius of curvature (C{meter}), M{a**2 / b}. @see: U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>} and U{Moritz, H. (1980), Geodetic Reference System 1980 <https://WikiPedia.org/wiki/Earth_radius#cite_note-Moritz-2>}. @note: Symbol C{c} is used by IUGG and IERS for the U{polar radius of curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}, see L{c2} and L{R2} or L{Rauthalic}. ''' return Radius(a2_b=self.a2 / self.b if self.f else self.a) # = rocPolar @Property_RO def a2_b2(self): '''Get the ratio I{equatorial} over I{polar} radius I{squared} (C{float}), M{(a / b)**2} == M{1 / (1 - e**2)} == M{1 / (1 - e2)} == M{1 / e21}. ''' return Float(a2_b2=self.a_b**2 if self.f else _1_0) @Property_RO def a_f(self): '''Get the I{equatorial} radius and I{flattening} (L{a_f2Tuple}), see method C{toEllipsoid2}. ''' return a_f2Tuple(self.a, self.f, name=self.name) @Property_RO def A(self): '''Get the UTM I{meridional (or rectifying)} radius (C{meter}). @see: I{Meridian arc unit} U{Q<https://StudyLib.net/doc/7443565/ a-highly-accurate-world-wide-algorithm-for-the-transverse...>}. ''' A, n = self.a, self.n if n: d = (n + _1_0) * 1048576 / A if d: # use 6 n**2 terms, half-way between the _KsOrder's 4, 6, 8 # <https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html> # <https://GeographicLib.SourceForge.io/C++/doc/transversemercator.html> and # <https://www.MyGeodesy.id.AU/documents/Karney-Krueger%20equations.pdf> (3) # A *= fhorner(n**2, 1048576, 262144, 16384, 4096, 1600, 784, 441) / 1048576) / (1 + n) A = Radius(A=Fhorner(n**2, 1048576, 262144, 16384, 4096, 1600, 784, 441).fover(d)) return A @Property_RO def _albersCyl(self): '''(INTERNAL) Helper for C{auxAuthalic}. ''' return _MODS.albers.AlbersEqualAreaCylindrical(datum=self, name=self.name) @Property_RO def AlphaKs(self): '''Get the I{Krüger} U{Alpha series coefficients<https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>} (C{KsOrder}C{-tuple}). ''' return self._Kseries( # XXX int/int quotients may require from __future__ import division as _; del _ # PYCHOK semicolon # n n**2 n**3 n**4 n**5 n**6 n**7 n**8 _T(1/2, -2/3, 5/16, 41/180, -127/288, 7891/37800, 72161/387072, -18975107/50803200), _T(13/48, -3/5, 557/1440, 281/630, -1983433/1935360, 13769/28800, 148003883/174182400), # PYCHOK unaligned _T(61/240, -103/140, 15061/26880, 167603/181440, -67102379/29030400, 79682431/79833600), # PYCHOK unaligned _T(49561/161280, -179/168, 6601661/7257600, 97445/49896, -40176129013/7664025600), # PYCHOK unaligned _T(34729/80640, -3418889/1995840, 14644087/9123840, 2605413599/622702080), # PYCHOK unaligned _T(212378941/319334400, -30705481/10378368, 175214326799/58118860800), # PYCHOK unaligned _T(1522256789/1383782400, -16759934899/3113510400), # PYCHOK unaligned _T(1424729850961/743921418240)) # PYCHOK unaligned @Property_RO def area(self): '''Get the ellipsoid's surface area (C{meter} I{squared}), M{4 * PI * c2}. @see: Properties L{areax}, L{c2} and L{R2} and functions L{ellipsoidalExact.areaOf} and L{ellipsoidalKarney.areaOf}. ''' return Meter2(area=self.c2 * PI4) @Property_RO def areax(self): '''Get the ellipsoid's surface area (C{meter} I{squared}), M{4 * PI * c2x}, more accurate for very I{oblate} ellipsoids. @see: Properties L{area}, L{c2x} and L{R2x}, class L{GeodesicExact} and functions L{ellipsoidalExact.areaOf} and L{ellipsoidalKarney.areaOf}. ''' return Meter2(areax=self.c2x * PI4) def _assert(self, val, eps=_TOL, f0=_0_0, Error=_AssertionError, txt=NN, **name_value): '''(INTERNAL) Assert a C{name=value} vs C{val}. ''' for n, v in name_value.items(): if abs(v - val) > eps: # PYCHOK no cover t = (v, _vs_, val) t = _SPACE_.join(strs(t, prec=12, fmt=Fmt.g)) t = Fmt.EQUAL(self._DOT_(n), t) raise Error(t, txt=txt or Fmt.exceeds_eps(eps)) return Float(v if self.f else f0, name=n) raise Error(unstr(self._DOT_(self._assert.__name__), val, eps=eps, f0=f0, **name_value)) def auxAuthalic(self, lat, inverse=False): '''Compute the I{authalic} auxiliary latitude or the I{inverse} thereof. @arg lat: The geodetic (or I{authalic}) latitude (C{degrees90}). @kwarg inverse: If C{True}, B{C{lat}} is the I{authalic} and return the geodetic latitude (C{bool}). @return: The I{authalic} (or geodetic) latitude in C{degrees90}. @see: U{Inverse-/AuthalicLatitude<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>}, U{Authalic latitude <https://WikiPedia.org/wiki/Latitude#Authalic_latitude>}, and U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, p 16. ''' if self.f: f = self._albersCyl._tanf if inverse else self._albersCyl._txif # PYCHOK attr lat = atand(f(tan(Phi_(lat)))) # PYCHOK attr return _aux(lat, inverse, Ellipsoid.auxAuthalic) def auxConformal(self, lat, inverse=False): '''Compute the I{conformal} auxiliary latitude or the I{inverse} thereof. @arg lat: The geodetic (or I{conformal}) latitude (C{degrees90}). @kwarg inverse: If C{True}, B{C{lat}} is the I{conformal} and return the geodetic latitude (C{bool}). @return: The I{conformal} (or geodetic) latitude in C{degrees90}. @see: U{Inverse-/ConformalLatitude<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>}, U{Conformal latitude <https://WikiPedia.org/wiki/Latitude#Conformal_latitude>}, and U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 15-16. ''' if self.f: f = self.es_tauf if inverse else self.es_taupf # PYCHOK attr lat = atand(f(tan(Phi_(lat)))) # PYCHOK attr return _aux(lat, inverse, Ellipsoid.auxConformal) def auxGeocentric(self, lat, inverse=False): '''Compute the I{geocentric} auxiliary latitude or the I{inverse} thereof. @arg lat: The geodetic (or I{geocentric}) latitude (C{degrees90}). @kwarg inverse: If C{True}, B{C{lat}} is the geocentric and return the I{geocentric} latitude (C{bool}). @return: The I{geocentric} (or geodetic) latitude in C{degrees90}. @see: U{Inverse-/GeocentricLatitude<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>}, U{Geocentric latitude <https://WikiPedia.org/wiki/Latitude#Geocentric_latitude>}, and U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 17-18. ''' if self.f: f = self.a2_b2 if inverse else self.b2_a2 lat = atand(f * tan(Phi_(lat))) return _aux(lat, inverse, Ellipsoid.auxGeocentric) def auxIsometric(self, lat, inverse=False): '''Compute the I{isometric} auxiliary latitude or the I{inverse} thereof. @arg lat: The geodetic (or I{isometric}) latitude (C{degrees}). @kwarg inverse: If C{True}, B{C{lat}} is the I{isometric} and return the geodetic latitude (C{bool}). @return: The I{isometric} (or geodetic) latitude in C{degrees}. @note: The I{isometric} latitude for geodetic C{+/-90} is far outside the C{[-90..+90]} range but the inverse thereof is the original geodetic latitude. @see: U{Inverse-/IsometricLatitude<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>}, U{Isometric latitude <https://WikiPedia.org/wiki/Latitude#Isometric_latitude>}, and U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 15-16. ''' if self.f: r = Phi_(lat, clip=0) lat = degrees(atan(self.es_tauf(sinh(r))) if inverse else asinh(self.es_taupf(tan(r)))) # clip=0, since auxIsometric(+/-90) is far outside [-90..+90] return _aux(lat, inverse, Ellipsoid.auxIsometric, clip=0) def auxParametric(self, lat, inverse=False): '''Compute the I{parametric} auxiliary latitude or the I{inverse} thereof. @arg lat: The geodetic (or I{parametric}) latitude (C{degrees90}). @kwarg inverse: If C{True}, B{C{lat}} is the I{parametric} and return the geodetic latitude (C{bool}). @return: The I{parametric} (or geodetic) latitude in C{degrees90}. @see: U{Inverse-/ParametricLatitude<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>}, U{Parametric latitude <https://WikiPedia.org/wiki/Latitude#Parametric_(or_reduced)_latitude>}, and U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, p 18. ''' if self.f: lat = self._beta(Lat(lat), inverse=inverse) return _aux(lat, inverse, Ellipsoid.auxParametric) auxReduced = auxParametric # synonymous def auxRectifying(self, lat, inverse=False): '''Compute the I{rectifying} auxiliary latitude or the I{inverse} thereof. @arg lat: The geodetic (or I{rectifying}) latitude (C{degrees90}). @kwarg inverse: If C{True}, B{C{lat}} is the I{rectifying} and return the geodetic latitude (C{bool}). @return: The I{rectifying} (or geodetic) latitude in C{degrees90}. @see: U{Inverse-/RectifyingLatitude<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>}, U{Rectifying latitude <https://WikiPedia.org/wiki/Latitude#Rectifying_latitude>}, and U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 16-17. ''' if self.f: lat = Lat(lat) if 0 < abs(lat) < _90_0: if inverse: e = self._elliptic_e22 d = degrees90(e.fEinv(e.cE * lat / _90_0)) lat = self.auxParametric(d, inverse=True) else: lat = _90_0 * self.Llat(lat) / self.L return _aux(lat, inverse, Ellipsoid.auxRectifying) @Property_RO def b(self): '''Get the I{polar} radius, semi-axis (C{meter}). ''' return self._b polaradius = b # = Rpolar @Property_RO def b_a(self): '''Get the ratio I{polar} over I{equatorial} radius (C{float}), M{b / a == f1 == 1 - f}. @see: Property L{f1}. ''' return self._assert(self.b / self.a, b_a=self.f1, f0=_1_0) @Property_RO def b2(self): '''Get the I{polar} radius I{squared} (C{float}), M{b**2}. ''' return Meter2(b2=self.b**2) @Property_RO def b2_a(self): '''Get the I{equatorial} meridional radius of curvature (C{meter}), M{b**2 / a}, see C{rocMeridional}C{(0)}. @see: U{Radii of Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. ''' return Radius(b2_a=self.b2 / self.a if self.f else self.b) @Property_RO def b2_a2(self): '''Get the ratio I{polar} over I{equatorial} radius I{squared} (C{float}), M{(b / a)**2} == M{(1 - f)**2} == M{1 - e**2} == C{e21}. ''' return Float(b2_a2=self.b_a**2 if self.f else _1_0) def _beta(self, lat, inverse=False): '''(INTERNAL) Get the I{parametric (or reduced) auxiliary latitude} or inverse thereof. ''' s, c = sincos2d(lat) # like Karney's tand(lat) s *= self.a_b if inverse else self.b_a return atan2d(s, c) # == atand(s / c) if c else copysign0(_90_0, lat) @Property_RO def BetaKs(self): '''Get the I{Krüger} U{Beta series coefficients<https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>} (C{KsOrder}C{-tuple}). ''' return self._Kseries( # XXX int/int quotients may require from __future__ import division as _; del _ # PYCHOK semicolon # n n**2 n**3 n**4 n**5 n**6 n**7 n**8 _T(1/2, -2/3, 37/96, -1/360, -81/512, 96199/604800, -5406467/38707200, 7944359/67737600), _T(1/48, 1/15, -437/1440, 46/105, -1118711/3870720, 51841/1209600, 24749483/348364800), # PYCHOK unaligned _T(17/480, -37/840, -209/4480, 5569/90720, 9261899/58060800, -6457463/17740800), # PYCHOK unaligned _T(4397/161280, -11/504, -830251/7257600, 466511/2494800, 324154477/7664025600), # PYCHOK unaligned _T(4583/161280, -108847/3991680, -8005831/63866880, 22894433/124540416), # PYCHOK unaligned _T(20648693/638668800, -16363163/518918400, -2204645983/12915302400), # PYCHOK unaligne _T(219941297/5535129600, -497323811/12454041600), # PYCHOK unaligned _T(191773887257/3719607091200)) # PYCHOK unaligned @deprecated_Property_RO def c(self): # PYCHOK no cover '''DEPRECATED, use property C{R2} or C{Rauthalic}.''' return self.R2 @Property_RO def c2(self): '''Get the I{authalic} earth radius I{squared} (C{meter} I{squared}). @see: Properties L{c2x}, L{area}, L{R2}, L{Rauthalic}, I{Karney's} U{equation (60) <https://Link.Springer.com/article/10.1007%2Fs00190-012-0578-z>} and C++ U{Ellipsoid.Area <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Authalic radius<https://WikiPedia.org/wiki/Earth_radius#Authalic_radius>}, U{Surface area <https://WikiPedia.org/wiki/Ellipsoid>} and U{surface area <https://www.Numericana.com/answer/geometry.htm#oblate>}. ''' return self._c2f(False) @Property_RO def c2x(self): '''Get the I{authalic} earth radius I{squared} (C{meter} I{squared}), more accurate for very I{oblate} ellipsoids. @see: Properties L{c2}, L{areax}, L{R2x}, L{Rauthalicx}, class L{GeodesicExact} and I{Karney}'s comments at C++ attribute U{GeodesicExact._c2<https://GeographicLib.SourceForge.io/C++/doc/GeodesicExact_8cpp_source.html>}. ''' return self._c2f(True) def _c2f(self, c2x): '''(INTERNAL) Helper for C{.c2} and C{.c2x}. ''' f, c2 = self.f, self.b2 if f: e = self.e if e > EPS0: if f > 0: # .isOblate c2 *= (asinh(sqrt(self.e22abs)) if c2x else atanh(e)) / e elif f < 0: # .isProlate c2 *= atan(e) / e # XXX asin? c2 = Meter2(c2=(self.a2 + c2) * _0_5) return c2 def circle4(self, lat): '''Get the equatorial or a parallel I{circle of latitude}. @arg lat: Geodetic latitude (C{degrees90}, C{str}). @return: A L{Circle4Tuple}C{(radius, height, lat, beta)} instance. @raise RangeError: Latitude B{C{lat}} outside valid range and L{pygeodesy.rangerrors} set to C{True}. @raise TypeError: Invalid B{C{lat}}. @raise ValueError: Invalid B{C{lat}}. @see: Definition of U{I{p} and I{z} under B{Parametric (or reduced) latitude} <https://WikiPedia.org/wiki/Latitude>}, I{Karney's} C++ U{CircleRadius and CircleHeight <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Ellipsoid.html>} and method C{Rlat}. ''' lat = Lat(lat) if lat: b = lat if abs(lat) < _90_0: if self.f: b = self._beta(lat) z, r = sincos2d(b) r *= self.a z *= self.b else: # near-polar r, z = _0_0, copysign0(self.b, lat) else: # equator r = self.a z = lat = b = _0_0 return Circle4Tuple(r, z, lat, b) def degrees2m(self, deg, lat=0): '''Convert an angle to the distance along the equator or along a parallel of (geodetic) latitude. @arg deg: The angle (C{degrees}). @kwarg lat: Parallel latitude (C{degrees90}, C{str}). @return: Distance (C{meter}, same units as the equatorial and polar radii) or C{0} for near-polar B{C{lat}}. @raise RangeError: Latitude B{C{lat}} outside valid range and L{pygeodesy.rangerrors} set to C{True}. @raise ValueError: Invalid B{C{deg}} or B{C{lat}}. ''' return self.radians2m(radians(deg), lat=lat) def distance2(self, lat0, lon0, lat1, lon1): '''I{Approximate} the distance and (initial) bearing between two points based on the U{local, flat earth approximation <https://www.EdWilliams.org/avform.htm#flat>} aka U{Hubeny <https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula. I{Suitable only for distances of several hundred Km or Miles and only between points not near-polar}. @arg lat0: From latitude (C{degrees}). @arg lon0: From longitude (C{degrees}). @arg lat1: To latitude (C{degrees}). @arg lon1: To longitude (C{degrees}). @return: A L{Distance2Tuple}C{(distance, initial)} with C{distance} in same units as this ellipsoid's axes. @note: The meridional and prime_vertical radii of curvature are taken and scaled I{at the initial latitude}, see C{roc2}. @see: Function L{pygeodesy.flatLocal}/L{pygeodesy.hubeny}. ''' phi0 = Phi_(lat0=lat0) m, n = self.roc2_(phi0, scaled=True) m *= Phi_(lat1=lat1) - phi0 n *= Lam_(lon1=lon1) - Lam_(lon0=lon0) return Distance2Tuple(hypot(m, n), atan2b(n, m)) @Property_RO def e(self): '''Get the I{unsigned, (1st) eccentricity} (C{float}), M{sqrt(1 - (b / a)**2))}, see C{a_b2e}. @see: Property L{es}. ''' return Float(e=sqrt(self.e2abs) if self.e2 else _0_0) @deprecated_Property_RO def e12(self): # see property ._e12 '''DEPRECATED, use property C{e21}.''' return self.e21 # @Property_RO # def _e12(self): # see property ._elliptic_e12 # # (INTERNAL) until e12 above can be replaced with e21. # return self.e2 / (_1_0 - self.e2) # see I{Karney}'s Ellipsoid._e12 = e2 / (1 - e2) @Property_RO def e2(self): '''Get the I{signed, (1st) eccentricity squared} (C{float}), M{f * (2 - f) == 1 - (b / a)**2}, see C{a_b2e2}. ''' return self._assert(a_b2e2(self.a, self.b), e2=f2e2(self.f)) @Property_RO def e2abs(self): '''Get the I{unsigned, (1st) eccentricity squared} (C{float}). ''' return abs(self.e2) @Property_RO def e21(self): '''Get 1 less I{1st eccentricity squared} (C{float}), M{1 - e**2} == M{1 - e2} == M{(1 - f)**2} == M{b**2 / a**2}, see C{b2_a2}. ''' return self._assert((_1_0 - self.f)**2, e21=_1_0 - self.e2, f0=_1_0) # _e2m = e21 # see I{Karney}'s Ellipsoid._e2m = 1 - _e2 _1_e21 = a2_b2 # == M{1 / e21} == M{1 / (1 - e**2)} @Property_RO def e22(self): '''Get the I{signed, 2nd eccentricity squared} (C{float}), M{e2 / (1 - e2) == e2 / (1 - f)**2 == (a / b)**2 - 1}, see C{a_b2e22}. ''' return self._assert(a_b2e22(self.a, self.b), e22=f2e22(self.f)) @Property_RO def e22abs(self): '''Get the I{unsigned, 2nd eccentricity squared} (C{float}). ''' return abs(self.e22) @Property_RO def e32(self): '''Get the I{signed, 3rd eccentricity squared} (C{float}), M{e2 / (2 - e2) == (a**2 - b**2) / (a**2 + b**2)}, see C{a_b2e32}. ''' return self._assert(a_b2e32(self.a, self.b), e32=f2e32(self.f)) @Property_RO def e32abs(self): '''Get the I{unsigned, 3rd eccentricity squared} (C{float}). ''' return abs(self.e32) @Property_RO def e4(self): '''Get the I{unsignd, (1st) eccentricity} to 4th power (C{float}), M{e**4 == e2**2}. ''' return Float(e4=self.e2**2 if self.e2 else _0_0) eccentricity = e # eccentricity # eccentricity2 = e2 # eccentricity squared eccentricity1st2 = e2 # first eccentricity squared eccentricity2nd2 = e22 # second eccentricity squared eccentricity3rd2 = e32 # third eccentricity squared def ecef(self, Ecef=None): '''Return U{ECEF<https://WikiPedia.org/wiki/ECEF>} converter. @kwarg Ecef: ECEF class to use, default L{EcefKarney}. @return: An ECEF converter for this C{ellipsoid}. @raise TypeError: Invalid B{C{Ecef}}. @see: Module L{pygeodesy.ecef}. ''' return _MODS.ecef._4Ecef(self, Ecef) @Property_RO def _elliptic_e12(self): # see I{Karney}'s Ellipsoid._e12 '''(INTERNAL) Elliptic helper for C{rhumbx.Rhumb}. ''' e12 = self.e2 / (_1_0 - self.e2) # NOT DEPRECATED .e12! return _MODS.elliptic.Elliptic(-e12) @Property_RO def _elliptic_e22(self): # aka ._elliptic_ep2 '''(INTERNAL) Elliptic helper for C{auxRectifying}, C{L}, C{Llat}. ''' return _MODS.elliptic.Elliptic(-self.e22abs) # complex equatoradius = a # Requatorial def e2s(self, s): '''Compute norm M{sqrt(1 - e2 * s**2)}. @arg s: Sine value (C{scalar}). @return: Norm (C{float}). @raise ValueError: Invalid B{C{s}}. ''' return sqrt(self.e2s2(s)) if self.e2 else _1_0 def e2s2(self, s): '''Compute M{1 - e2 * s**2}. @arg s: Sine value (C{scalar}). @return: Result (C{float}). @raise ValueError: Invalid B{C{s}}. ''' r = _1_0 if self.e2: try: r -= self.e2 * Scalar(s=s)**2 if r < 0: raise ValueError(_negative_) except (TypeError, ValueError) as x: t = self._DOT_(Ellipsoid.e2s2.__name__) raise _ValueError(t, s, cause=x) return r @Property_RO def es(self): '''Get the I{signed (1st) eccentricity} (C{float}). @see: Property L{e}. ''' # note, self.e is always non-negative return Float(es=copysign0(self.e, self.f)) # see .ups def es_atanh(self, x): '''Compute M{es * atanh(es * x)} or M{-es * atan(es * x)} for I{oblate} respectively I{prolate} ellipsoids where I{es} is the I{signed} (1st) eccentricity. @raise ValueError: Invalid B{C{x}}. @see: Function U{Math::eatanhe<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Math.html>}. ''' return self._es_atanh(Scalar(x=x)) if self.f else _0_0 def _es_atanh(self, x): '''(INTERNAL) Helper for .es_atanh, ._es_taupf2 and ._exp_es_atanh. ''' es = self.es # signOf(es) == signOf(f) return es * (atanh(es * x) if es > 0 else # .isOblate (-atan(es * x) if es < 0 else # .isProlate _0_0)) # .isSpherical @Property_RO def es_c(self): '''Get M{(1 - f) * exp(es_atanh(1))} (C{float}), M{b_a * exp(es_atanh(1))}. ''' return Float(es_c=(self._exp_es_atanh_1 * self.b_a) if self.f else _1_0) def es_tauf(self, taup): '''Compute I{Karney}'s U{equations (19), (20) and (21) <https://ArXiv.org/abs/1002.1417>}. @see: I{Karney}'s C++ method U{Math::tauf<https://GeographicLib. SourceForge.io/html/classGeographicLib_1_1Math.html>} and and I{Veness}' JavaScript method U{toLatLon<https://www. Movable-Type.co.UK/scripts/latlong-utm-mgrs.html>}. ''' t = Scalar(taup=taup) if self.f: # .isEllipsoidal a = abs(t) T = t * (self._exp_es_atanh_1 if a > 70 else self._1_e21) if abs(T * _EPSqrt) < _2_0: # handles +/- INF and NAN s = (a * _TOL) if a > _1_0 else _TOL for T, _, d in self._es_tauf3(t, T): # max 2 if abs(d) < s: break t = Scalar(tauf=T) return t def _es_tauf3(self, taup, T, N=9): # in .utm.Utm._toLLEB '''(INTERNAL) Yield a 3-tuple C{(τi, iteration, delta)} for at most B{C{N}} Newton iterations, converging rapidly except when C{delta} toggles on +/-1.12e-16 or +/-4.47e-16, see C{.utm.Utm._toLLEB}. ''' e = self._1_e21 _F2_ = Fsum(T).fsum2_ # τ0 _tf2 = self._es_taupf2 for i in range(1, N + 1): a, h = _tf2(T) d = (taup - a) * (e + T**2) / (hypot1(a) * h) # = (taup - a) / hypot1(a) / ((e + T**2) / h) T, d = _F2_(d) # τi, (τi - τi-1) yield T, i, d def es_taupf(self, tau): '''Compute I{Karney}'s U{equations (7), (8) and (9) <https://ArXiv.org/abs/1002.1417>}. @see: I{Karney}'s C++ method U{Math::taupf<https://GeographicLib. SourceForge.io/html/classGeographicLib_1_1Math.html>}. ''' t = Scalar(tau=tau) if self.f: # .isEllipsoidal t, _ = self._es_taupf2(t) t = Scalar(taupf=t) return t def _es_taupf2(self, tau): '''(INTERNAL) Return 2-tuple C{(es_taupf(tau), hypot1(tau))}. ''' if _isfinite(tau): h = hypot1(tau) s = sinh(self._es_atanh(tau / h)) a = hypot1(s) * tau - h * s else: a, h = tau, INF return a, h @Property_RO def _exp_es_atanh_1(self): '''(INTERNAL) Helper for .es_c and .es_tauf. ''' return exp(self._es_atanh(_1_0)) if self.es else _1_0 @Property_RO def f(self): '''Get the I{flattening} (C{float}), M{(a - b) / a}, C{0} for spherical, negative for prolate. ''' return self._f @Property_RO def f_(self): '''Get the I{inverse flattening} (C{float}), M{1 / f} == M{a / (a - b)}, C{0} for spherical, see C{a_b2f_}. ''' return self._f_ @Property_RO def f1(self): '''Get the I{1 - flattening} (C{float}), M{f1 == 1 - f == b / a}. @see: Property L{b_a}. ''' return Float(f1=_1_0 - self.f) @Property_RO def f2(self): '''Get the I{2nd flattening} (C{float}), M{(a - b) / b == f / (1 - f)}, C{0} for spherical, see C{a_b2f2}. ''' return self._assert(self.a_b - _1_0, f2=f2f2(self.f)) @Property_RO def geodesic(self): '''Get this ellipsoid's I{wrapped Karney} U{Geodesic <https://GeographicLib.SourceForge.io/C++/doc/python/code.html>}, provided the U{geographiclib<https://PyPI.org/project/geographiclib>} package is installed. ''' # if not self.isEllipsoidal: # raise _IsnotError(_ellipsoidal_, ellipsoid=self) return _MODS.karney._wrapped.Geodesic(self) @Property_RO def geodesicx(self): '''Get this ellipsoid's L{GeodesicExact}. ''' # if not self.isEllipsoidal: # raise _IsnotError(_ellipsoidal_, ellipsoid=self) return _MODS.geodesicx.GeodesicExact(self, name=self.name) @Property_RO def geodsolve(self): '''Get this ellipsoid's L{GeodesicSolve}, the I{wrapper} around utility U{GeodSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>}, provided the path to the C{GeodSolve} executable is specified with env variable C{PYGEODESY_GEODSOLVE}. ''' # if not self.isEllipsoidal: # raise _IsnotError(_ellipsoidal_, ellipsoid=self) return _MODS.geodsolve.GeodesicSolve(self, name=self.name) def height4(self, xyz, normal=True): '''Compute the projection on and the height of a cartesian above or below this ellipsoid's surface. @arg xyz: The cartesian (C{Cartesian}, L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}). @kwarg normal: If C{True} the projection is perpendicular to (the nearest point on) this ellipsoid's surface, otherwise the C{radial} line to this ellipsoid's center (C{bool}). @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates C{x}, C{y} and C{z} of the projection on or the intersection with and with the height C{h} above the ellipsoid's surface in C{meter}, conventionally. @raise ValueError: Null B{C{xyz}}. @raise TypeError: Non-cartesian B{C{xyz}}. @see: U{Distance to<https://StackOverflow.com/questions/22959698/distance-from-given-point-to-given-ellipse>} and U{intersection with<https://MathWorld.wolfram.com/Ellipse-LineIntersection.html>} an ellipse and method L{Triaxial.height4}. ''' v = _MODS.vector3d._otherV3d(xyz=xyz) r = v.length a, b, i = self.a, self.b, None if r < EPS0: # EPS v = v.times(_0_0) h = -a elif self.isSpherical: v = v.times(a / r) h = r - a elif normal: # perpendicular to ellipsoid x, y = hypot(v.x, v.y), abs(v.z) if x < EPS0: # PYCHOK no cover z = copysign0(b, v.z) v = Vector3Tuple(v.x, v.y, z) h = y - b # polar elif y < EPS0: # PYCHOK no cover t = a / r v = v.times_(t, t, 0) # force z=0.0 h = x - a # equatorial else: # normal in 1st quadrant x, y, i = _normalTo3(x, y, self) t, v = v, v.times_(x, x, y) h = t.minus(v).length else: # radial to ellipsoid's center h = hypot_(a * v.z, b * v.x, b * v.y) t = (a * b / h) if h > EPS0 else _0_0 # EPS v = v.times(t) h = r * (_1_0 - t) return Vector4Tuple(v.x, v.y, v.z, h, iteration=i, name=self.height4.__name__) def _hubeny_2(self, phi2, phi1, lam21): '''(INTERNAL) like function C{pygeodesy.flatLocal_}/C{pygeodesy.hubeny_} but returning the I{angular} distance in C{radians squared}. ''' m, n = self.roc2_((phi2 + phi1) * _0_5, scaled=True) return hypot2(m * (phi2 - phi1), n * lam21) * self.a2_ @Property_RO def isEllipsoidal(self): '''Is this model I{ellipsoidal} (C{bool})? ''' return self.f != 0 @Property_RO def isOblate(self): '''Is this ellipsoid I{oblate} (C{bool})? I{Prolate} or spherical otherwise. ''' return self.f > 0 @Property_RO def isProlate(self): '''Is this ellipsoid I{prolate} (C{bool})? I{Oblate} or spherical otherwise. ''' return self.f < 0 @Property_RO def isSpherical(self): '''Is this ellipsoid I{spherical} (C{bool})? ''' return self.f == 0 def _Kseries(self, *AB8Ks): '''(INTERNAL) Compute the 4-, 6- or 8-th order I{Krüger} Alpha or Beta series coefficients per I{Karney}'s U{equations (35) and (36)<https://ArXiv.org/pdf/1002.1417v3.pdf>}. @arg AB8Ks: 8-Tuple of 8-th order I{Krüger} Alpha or Beta series coefficient tuples. @return: I{Krüger} series coefficients (L{KsOrder}C{-tuple}). @see: I{Karney}'s 30-th order U{TMseries30 <https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>}. ''' k = self.KsOrder if self.n: ns = fpowers(self.n, k) ks = tuple(fdot(AB8Ks[i][:k-i], *ns[i:]) for i in range(k)) else: ks = _0_0s(k) return ks @property_doc_(''' the I{Krüger} series' order (C{int}), see properties C{AlphaKs}, C{BetaKs}.''') def KsOrder(self): '''Get the I{Krüger} series' order (C{int} 4, 6 or 8). ''' return self._KsOrder @KsOrder.setter # PYCHOK setter! def KsOrder(self, order): '''Set the I{Krüger} series' order (C{int} 4, 6 or 8). @raise ValueError: Invalid B{C{order}}. ''' if not (isint(order) and order in (4, 6, 8)): raise _ValueError(order=order) if order != self._KsOrder: Ellipsoid.AlphaKs._update(self) Ellipsoid.BetaKs._update(self) self._KsOrder = order @Property_RO def L(self): '''Get the I{quarter meridian} C{L}, aka the C{polar distance} along a meridian between the equator and a pole (C{meter}), M{b * Elliptic(-e2 / (1 - e2)).E} or M{b * PI / 2}. ''' r = self._elliptic_e22.cE if self.f else PI_2 return Distance(L=self.b * r) @Property_RO def _L_90(self): '''Get the I{quarter meridian} per degree (C{meter}), M{self.L / 90}. ''' return self.L / _90_0 def Llat(self, lat): '''Return the I{meridional length}, the distance along a meridian between the equator and a (geodetic) latitude, see C{L}. @arg lat: Geodetic latitude (C{degrees90}). @return: The meridional length at B{C{lat}}, negative on southern hemisphere (C{meter}). ''' r = self._elliptic_e22.fEd(self.auxParametric(lat)) if self.f else Phi_(lat) return Distance(Llat=self.b * r) Lmeridian = Llat # meridional distance @deprecated_Property_RO def majoradius(self): # PYCHOK no cover '''DEPRECATED, use property C{a} or C{Requatorial}.''' return self.a def m2degrees(self, distance, lat=0): '''Convert a distance to an angle along the equator or along a parallel of (geodetic) latitude. @arg distance: Distance (C{meter}). @kwarg lat: Parallel latitude (C{degrees90}, C{str}). @return: Angle (C{degrees}) or C{INF} for near-polar B{C{lat}}. @raise RangeError: Latitude B{C{lat}} outside valid range and L{pygeodesy.rangerrors} set to C{True}. @raise ValueError: Invalid B{C{distance}} or B{C{lat}}. ''' return degrees(self.m2radians(distance, lat=lat)) def m2radians(self, distance, lat=0): '''Convert a distance to an angle along the equator or along a parallel of (geodetic) latitude. @arg distance: Distance (C{meter}). @kwarg lat: Parallel latitude (C{degrees90}, C{str}). @return: Angle (C{radians}) or C{INF} for near-polar B{C{lat}}. @raise RangeError: Latitude B{C{lat}} outside valid range and L{pygeodesy.rangerrors} set to C{True}. @raise ValueError: Invalid B{C{distance}} or B{C{lat}}. ''' r = self.circle4(lat).radius if lat else self.a return m2radians(distance, radius=r, lat=0) @deprecated_Property_RO def minoradius(self): # PYCHOK no cover '''DEPRECATED, use property C{b} or C{Rpolar}.''' return self.b @Property_RO def n(self): '''Get the I{3rd flattening} (C{float}), M{f / (2 - f) == (a - b) / (a + b)}, see C{a_b2n}. ''' return self._assert(a_b2n(self.a, self.b), n=f2n(self.f)) flattening = f flattening1st = f flattening2nd = f2 flattening3rd = n polaradius = b # Rpolar # @Property_RO # def Q(self): # '''Get the I{meridian arc unit} C{Q}, the mean, meridional length I{per radian} C({float}). # # @note: C{Q * PI / 2} ≈ C{L}, the I{quarter meridian}. # # @see: Property C{A} and U{Engsager, K., Poder, K.<https://StudyLib.net/doc/7443565/ # a-highly-accurate-world-wide-algorithm-for-the-transverse...>}. # ''' # n = self.n # d = (n + _1_0) / self.a # return Float(Q=Fhorner(n**2, _1_0, _0_25, _1_16th, _0_25).fover(d) if d else self.b) # # Moritz, H. <https://Geodesy.Geology.Ohio-State.EDU/course/refpapers/00740128.pdf> # # Q = (1 - 3/4 * e'2 + 45/64 * e'4 - 175/256 * e'6 + 11025/16384 * e'8) * rocPolar # # = (4 + e'2 * (-3 + e'2 * (45/16 + e'2 * (-175/64 + e'2 * 11025/4096)))) * rocPolar / 4 # return Fhorner(self.e22, 4, -3, 45 / 16, -175 / 64, 11025 / 4096).fover(4 / self.rocPolar) @deprecated_Property_RO def quarteradius(self): # PYCHOK no cover '''DEPRECATED, use property C{L} or method C{Llat}.''' return self.L @Property_RO def R1(self): '''Get the I{mean} earth radius per I{IUGG} (C{meter}), M{(2 * a + b) / 3 == a * (1 - f / 3)}. @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>} and method C{Rgeometric}. ''' r = Fsum(self.a, self.a, self.b).fover(_3_0) if self.f else self.a return Radius(R1=r) Rmean = R1 @Property_RO def R2(self): '''Get the I{authalic} earth radius (C{meter}), M{sqrt(c2)}. @see: C{R2x}, C{c2}, C{area} and U{Earth radius <https://WikiPedia.org/wiki/Earth_radius>}. ''' return Radius(R2=sqrt(self.c2) if self.f else self.a) Rauthalic = R2 # @Property_RO # def R2(self): # # Moritz, H. <https://Geodesy.Geology.Ohio-State.EDU/course/refpapers/00740128.pdf> # # R2 = (1 - 2/3 * e'2 + 26/45 * e'4 - 100/189 * e'6 + 7034/14175 * e'8) * rocPolar # # = (3 + e'2 * (-2 + e'2 * (26/15 + e'2 * (-100/63 + e'2 * 7034/4725)))) * rocPolar / 3 # return Fhorner(self.e22, 3, -2, 26 / 15, -100 / 63, 7034 / 4725).fover(3 / self.rocPolar) @Property_RO def R2x(self): '''Get the I{authalic} earth radius (C{meter}), M{sqrt(c2x)}. @see: C{R2}, C{c2x} and C{areax}. ''' return Radius(R2x=sqrt(self.c2x) if self.f else self.a) Rauthalicx = R2x @Property_RO def R3(self): '''Get the I{volumetric} earth radius (C{meter}), M{(a * a * b)**(1/3)}. @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>} and C{volume}. ''' r = (cbrt(self.b_a) * self.a) if self.f else self.a return Radius(R3=r) Rvolumetric = R3 def radians2m(self, rad, lat=0): '''Convert an angle to the distance along the equator or along a parallel of (geodetic) latitude. @arg rad: The angle (C{radians}). @kwarg lat: Parallel latitude (C{degrees90}, C{str}). @return: Distance (C{meter}, same units as the equatorial and polar radii) or C{0} for near-polar B{C{lat}}. @raise RangeError: Latitude B{C{lat}} outside valid range and L{pygeodesy.rangerrors} set to C{True}. @raise ValueError: Invalid B{C{rad}} or B{C{lat}}. ''' r = self.circle4(lat).radius if lat else self.a return radians2m(rad, radius=r, lat=0) @Property_RO def Rbiaxial(self): '''Get the I{biaxial, quadratic} mean earth radius (C{meter}), M{sqrt((a**2 + b**2) / 2)}. @see: C{Rtriaxial} ''' a, b = self.a, self.b q = (sqrt((_1_0 + self.b2_a2) * _0_5) * a) if a > b else ( (sqrt((_1_0 + self.a2_b2) * _0_5) * b) if a < b else a) return Radius(Rbiaxial=q) Requatorial = a # for consistent naming def Rgeocentric(self, lat): '''Compute the I{geocentric} earth radius of (geodetic) latitude. @arg lat: Latitude (C{degrees90}). @return: Geocentric earth radius (C{meter}). @raise ValueError: Invalid B{C{lat}}. @see: U{Geocentric Radius <https://WikiPedia.org/wiki/Earth_radius#Geocentric_radius>} ''' r, a = self.a, Phi_(lat) if a and self.f: if abs(a) < PI_2: s2, c2 = _s2_c2(a) b2_a2_s2 = self.b2_a2 * s2 # R == sqrt((a2**2 * c2 + b2**2 * s2) / (a2 * c2 + b2 * s2)) # == sqrt(a2**2 * (c2 + (b2 / a2)**2 * s2) / (a2 * (c2 + b2 / a2 * s2))) # == sqrt(a2 * (c2 + (b2 / a2)**2 * s2) / (c2 + (b2 / a2) * s2)) # == a * sqrt((c2 + b2_a2 * b2_a2 * s2) / (c2 + b2_a2 * s2)) # == a * sqrt((c2 + b2_a2 * b2_a2_s2) / (c2 + b2_a2_s2)) r *= sqrt((c2 + b2_a2_s2 * self.b2_a2) / (c2 + b2_a2_s2)) else: r = self.b return Radius(Rgeocentric=r) @Property_RO def Rgeometric(self): '''Get the I{geometric} mean earth radius (C{meter}), M{sqrt(a * b)}. @see: C{R1}. ''' g = sqrt(self.a * self.b) if self.f else self.a return Radius(Rgeometric=g) @Property_RO def rhumbsolve(self): '''Get this ellipsoid's L{RhumbSolve}, the I{wrapper} around utility U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>}, provided the path to the C{RhumbSolve} executable is specified with env variable C{PYGEODESY_RHUMBSOLVE}. ''' # if not self.isEllipsoidal: # raise _IsnotError(_ellipsoidal_, ellipsoid=self) return _MODS.rhumbsolve.RhumbSolve(self, name=self.name) @Property_RO def rhumbx(self): '''Get this ellipsoid's L{Rhumb}. ''' # if not self.isEllipsoidal: # raise _IsnotError(_ellipsoidal_, ellipsoid=self) return _MODS.rhumbx.Rhumb(self, name=self.name) def Rlat(self, lat): '''I{Approximate} the earth radius of (geodetic) latitude. @arg lat: Latitude (C{degrees90}). @return: Approximate earth radius (C{meter}). @raise RangeError: Latitude B{C{lat}} outside valid range and L{pygeodesy.rangerrors} set to C{True}. @raise TypeError: Invalid B{C{lat}}. @raise ValueError: Invalid B{C{lat}}. @note: C{Rlat(B{90})} equals C{Rpolar}. @see: Method C{circle4}. ''' # r = a - (a - b) * |lat| / 90 r = self.a if self.f and lat: # .isEllipsoidal r -= (r - self.b) * abs(Lat(lat)) / _90_0 r = Radius(Rlat=r) return r Rpolar = b # for consistent naming @deprecated_Property_RO def Rquadratic(self): # PYCHOK no cover '''DEPRECATED, use property C{Rbiaxial} or C{Rtriaxial}.''' return self.Rbiaxial @deprecated_Property_RO def Rr(self): # PYCHOK no cover '''DEPRECATED, use property C{Rrectifying}.''' return self.Rrectifying @Property_RO def Rrectifying(self): '''Get the I{rectifying} earth radius (C{meter}), M{((a**(3/2) + b**(3/2)) / 2)**(2/3)}. @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}. ''' r = (cbrt2((_1_0 + sqrt3(self.b_a)) * _0_5) * self.a) if self.f else self.a return Radius(Rrectifying=r) def roc1_(self, sa, ca=None): '''Compute the I{prime-vertical}, I{normal} radius of curvature of (geodetic) latitude, I{unscaled}. @arg sa: Sine of the latitude (C{float}, [-1.0..+1.0]). @kwarg ca: Optional cosine of the latitude (C{float}, [-1.0..+1.0]) to use an alternate formula. @return: The prime-vertical radius of curvature (C{float}). @note: The delta between both formulae with C{Ellipsoids.WGS84} is less than 2 nanometer over the entire latitude range. @see: Method L{roc2_} and class L{EcefYou}. ''' if not self.f: # .isSpherical n = self.a elif ca is None: r = self.e2s2(sa) # see .roc2_ and _EcefBase._forward n = (self.a / sqrt(r)) if r > EPS02 else _0_0 elif ca: # derived from EcefYou.forward h = hypot(ca, self.b_a * sa) if sa else abs(ca) n = self.a / h elif sa: n = self.a2_b / abs(sa) else: n = self.a return n def roc2(self, lat, scaled=False): '''Compute the I{meridional} and I{prime-vertical}, I{normal} radii of curvature of (geodetic) latitude. @arg lat: Latitude (C{degrees90}). @kwarg scaled: Scale prime_vertical by C{cos(radians(B{lat}))} (C{bool}). @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with the radii of curvature. @raise ValueError: Invalid B{C{lat}}. @see: Methods L{roc2_} and L{roc1_}, U{Local, flat earth approximation <https://www.EdWilliams.org/avform.htm#flat>} and meridional and prime vertical U{Radii of Curvature<https://WikiPedia.org/wiki/ Earth_radius#Radii_of_curvature>}. ''' return self.roc2_(Phi_(lat), scaled=scaled) def roc2_(self, phi, scaled=False): '''Compute the I{meridional} and I{prime-vertical}, I{normal} radii of curvature of (geodetic) latitude. @arg phi: Latitude (C{radians}). @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}). @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with the radii of curvature. @raise ValueError: Invalid B{C{phi}}. @see: Methods L{roc2} and L{roc1_}, property L{rocEquatorial2}, U{Local, flat earth approximation<https://www.EdWilliams.org/avform.htm#flat>} and the meridional and prime vertical U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. ''' a = abs(Phi(phi)) if self.f: r = self.e2s2(sin(a)) if r > EPS02: n = self.a / sqrt(r) m = n * self.e21 / r # PYCHOK attr else: m = n = _0_0 # PYCHOK attr else: m = n = self.a if scaled and a: n *= cos(a) if a < PI_2 else _0_0 return Curvature2Tuple(Radius(rocMeridional=m), Radius(rocPrimeVertical=n)) def rocBearing(self, lat, bearing): '''Compute the I{directional} radius of curvature of (geodetic) latitude and compass direction. @arg lat: Latitude (C{degrees90}). @arg bearing: Direction (compass C{degrees360}). @return: Directional radius of curvature (C{meter}). @raise RangeError: Latitude B{C{lat}} outside valid range and L{pygeodesy.rangerrors} set to C{True}. @raise ValueError: Invalid B{C{lat}} or B{C{bearing}}. @see: U{Radii of Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>} ''' if self.f: s2, c2 = _s2_c2(Bearing_(bearing)) m, n = self.roc2_(Phi_(lat)) if n < m: # == n / (c2 * n / m + s2) c2 *= n / m elif m < n: # == m / (c2 + s2 * m / n) s2 *= m / n n = m b = n / (c2 + s2) # == 1 / (c2 / m + s2 / n) else: b = self.b # == self.a return Radius(rocBearing=b) @Property_RO def rocEquatorial2(self): '''Get the I{meridional} and I{prime-vertical}, I{normal} radii of curvature at the equator as L{Curvature2Tuple}C{(meridional, prime_vertical)}. @see: Methods L{rocMeridional} and L{rocPrimeVertical}, properties L{b2_a}, L{a2_b}, C{rocPolar} and polar and equatorial U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. ''' return Curvature2Tuple(Radius(rocMeridional0=self.b2_a if self.f else self.a), Radius(rocPrimeVertical0=self.a)) def rocGauss(self, lat): '''Compute the I{Gaussian} radius of curvature of (geodetic) latitude. @arg lat: Latitude (C{degrees90}). @return: Gaussian radius of curvature (C{meter}). @raise ValueError: Invalid B{C{lat}}. @see: Non-directional U{Radii of Curvature<https://WikiPedia.org/wiki/ Earth_radius#Radii_of_curvature>} ''' # using ... # m, n = self.roc2_(Phi_(lat)) # return sqrt(m * n) # ... requires 1 or 2 sqrt g = self.b if self.f: s2, c2 = _s2_c2(Phi_(lat)) g = g / (c2 + self.b2_a2 * s2) return Radius(rocGauss=g) def rocMean(self, lat): '''Compute the I{mean} radius of curvature of (geodetic) latitude. @arg lat: Latitude (C{degrees90}). @return: Mean radius of curvature (C{meter}). @raise ValueError: Invalid B{C{lat}}. @see: Non-directional U{Radii of Curvature<https://WikiPedia.org/wiki/ Earth_radius#Radii_of_curvature>} ''' if self.f: m, n = self.roc2_(Phi_(lat)) m *= _2_0 * n / (m + n) # == 2 / (1 / m + 1 / n) else: m = self.a return Radius(rocMean=m) def rocMeridional(self, lat): '''Compute the I{meridional} radius of curvature of (geodetic) latitude. @arg lat: Latitude (C{degrees90}). @return: Meridional radius of curvature (C{meter}). @raise ValueError: Invalid B{C{lat}}. @see: Methods L{roc2} and L{roc2_}, U{Local, flat earth approximation <https://www.EdWilliams.org/avform.htm#flat>} and U{Radii of Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. ''' return self.roc2_(Phi_(lat)).meridional if lat else \ self.rocEquatorial2.meridional rocPolar = a2_b # synonymous def rocPrimeVertical(self, lat): '''Compute the I{prime-vertical}, I{normal} radius of curvature of (geodetic) latitude, aka the I{transverse} radius of curvature. @arg lat: Latitude (C{degrees90}). @return: Prime-vertical radius of curvature (C{meter}). @raise ValueError: Invalid B{C{lat}}. @see: Methods L{roc2}, L{roc2_} and L{roc1_}, U{Local, flat earth approximation<https://www.EdWilliams.org/avform.htm#flat>} and U{Radii of Curvature<https://WikiPedia.org/wiki/ Earth_radius#Radii_of_curvature>}. ''' return self.roc2_(Phi_(lat)).prime_vertical if lat else \ self.rocEquatorial2.prime_vertical rocTransverse = rocPrimeVertical # synonymous @deprecated_Property_RO def Rs(self): # PYCHOK no cover '''DEPRECATED, use property C{Rgeometric}.''' return self.Rgeometric @Property_RO def Rtriaxial(self): '''Get the I{triaxial, quadratic} mean earth radius (C{meter}), M{sqrt((3 * a**2 + b**2) / 4)}. @see: C{Rbiaxial} ''' a, b = self.a, self.b q = (sqrt((_3_0 + self.b2_a2) * _0_25) * a) if a > b else ( (sqrt((_3_0 * self.a2_b2 + _1_0) * _0_25) * b) if a < b else a) return Radius(Rtriaxial=q) def toEllipsoid2(self, name=NN): '''Get a copy of this ellipsoid as an L{Ellipsoid2}. @kwarg name: Optional, unique name (C{str}). @see: Property C{a_f}. ''' return Ellipsoid2(self, None, name=name) def toStr(self, prec=8, name=NN, **unused): # PYCHOK expected '''Return this ellipsoid as a text string. @kwarg prec: Number of decimal digits, unstripped (C{int}). @kwarg name: Override name (C{str}) or C{None} to exclude this ellipsoid's name. @return: This C{Ellipsoid}'s attributes (C{str}). ''' E = Ellipsoid t = E.a, E.b, E.f_, E.f, E.f2, E.n, E.e, E.e2, E.e22, E.e32, \ E.A, E.L, E.R1, E.R2, E.R3, E.Rbiaxial, E.Rtriaxial return self._instr(name, prec, props=t) def toTriaxial(self, name=NN): '''Convert this ellipsoid to a L{Triaxial_}. @return: A L{Triaxial_} or L{Triaxial} with the C{X} axis pointing east and C{Z} pointing north. @see: Method L{Triaxial_.toEllipsoid}. ''' a, b, n = self.a, self.b, (name or self.name) return _MODS.triaxials.Triaxial_(a, a, b, name=n) if a < b else \ _MODS.triaxials.Triaxial( a, a, b, name=n) @Property_RO def volume(self): '''Get the ellipsoid's I{volume} (C{meter**3}), M{4 / 3 * PI * R3**3}. @see: C{R3}. ''' return Meter3(volume=self.a2 * self.b * PI_3 * _4_0) class Ellipsoid2(Ellipsoid): '''An L{Ellipsoid} specified by I{equatorial} radius and I{flattening}. ''' def __init__(self, a, f, name=NN): '''New L{Ellipsoid2}. @arg a: Equatorial radius, semi-axis (C{meter}). @arg f: Flattening: (C{float} < 1.0, negative for I{prolate}). @kwarg name: Optional, unique name (C{str}). @raise NameError: Ellipsoid with that B{C{name}} already exists. @raise ValueError: Invalid B{C{a}} or B{C{f}}. @note: C{abs(B{f}) < EPS} is forced to C{B{f}=0}, I{spherical}. Negative C{B{f}} produces a I{prolate} ellipsoid. ''' if f is None and isinstance(a, Ellipsoid): Ellipsoid.__init__(self, a.a, f =a.f, b=a.b, f_=a.f_, name=name) else: Ellipsoid.__init__(self, a, f=f, name=name) def _spherical_a_b(a, b): '''(INTERNAL) C{True} for spherical or invalid C{a} or C{b}. ''' return a < EPS or b < EPS or abs(a - b) < EPS def _spherical_f(f): '''(INTERNAL) C{True} for spherical or invalid C{f}. ''' return abs(f) < EPS or f > EPS1 def _spherical_f_(f_): '''(INTERNAL) C{True} for spherical or invalid C{f_}. ''' return abs(f_) < EPS or abs(f_) > _1_EPS def a_b2e(a, b): '''Return C{e}, the I{1st eccentricity} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The I{unsigned}, (1st) eccentricity (C{float} or C{0}), M{sqrt(1 - (b / a)**2)}. @note: The result is always I{non-negative} and C{0} for I{near-spherical} ellipsoids. ''' return Float(e=sqrt(abs(a_b2e2(a, b)))) def a_b2e2(a, b): '''Return C{e2}, the I{1st eccentricity squared} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The I{signed}, (1st) eccentricity I{squared} (C{float} or C{0}), M{1 - (b / a)**2}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. ''' return Float(e2=_0_0 if _spherical_a_b(a, b) else (_1_0 - (b / a)**2)) def a_b2e22(a, b): '''Return C{e22}, the I{2nd eccentricity squared} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The I{signed}, 2nd eccentricity I{squared} (C{float} or C{0}), M{(a / b)**2 - 1}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. ''' return Float(e22=_0_0 if _spherical_a_b(a, b) else ((a / b)**2 - _1_0)) def a_b2e32(a, b): '''Return C{e32}, the I{3rd eccentricity squared} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The I{signed}, 3rd eccentricity I{squared} (C{float} or C{0}), M{(a**2 - b**2) / (a**2 + b**2)}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. ''' a2, b2 = a**2, b**2 return Float(e32=_0_0 if _spherical_a_b(a2, b2) else ((a2 - b2) / (a2 + b2))) def a_b2f(a, b): '''Return C{f}, the I{flattening} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The flattening (C{float} or C{0}), M{(a - b) / a}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. ''' f = 0 if _spherical_a_b(a, b) else ((a - b) / a) return _f_0_0 if _spherical_f(f) else Float(f=f) def a_b2f_(a, b): '''Return C{f_}, the I{inverse flattening} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The inverse flattening (C{float} or C{0}), M{a / (a - b)}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. ''' f_ = 0 if _spherical_a_b(a, b) else (a / float(a - b)) return _f__0_0 if _spherical_f_(f_) else Float(f_=f_) def a_b2f2(a, b): '''Return C{f2}, the I{2nd flattening} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The I{signed}, 2nd flattening (C{float} or C{0}), M{(a - b) / b}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. ''' t = 0 if _spherical_a_b(a, b) else float(a - b) return Float(f2=_0_0 if abs(t) < EPS else (t / b)) def a_b2n(a, b): '''Return C{n}, the I{3rd flattening} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The I{signed}, 3rd flattening (C{float} or C{0}), M{(a - b) / (a + b)}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. ''' t = 0 if _spherical_a_b(a, b) else float(a - b) return Float(n=_0_0 if abs(t) < EPS else (t / (a + b))) def a_f2b(a, f): '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{flattening}. @arg a: Equatorial radius (C{scalar} > 0). @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @return: The polar radius (C{float}), M{a * (1 - f)}. ''' b = a if _spherical_f(f) else (a * (_1_0 - f)) return Radius_(b=a if _spherical_a_b(a, b) else b) def a_f_2b(a, f_): '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{inverse flattening}. @arg a: Equatorial radius (C{scalar} > 0). @arg f_: Inverse flattening (C{scalar} >>> 1). @return: The polar radius (C{float}), M{a * (f_ - 1) / f_}. ''' b = a if _spherical_f_(f_) else (a * (f_ - _1_0) / f_) return Radius_(b=a if _spherical_a_b(a, b) else b) def b_f2a(b, f): '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{flattening}. @arg b: Polar radius (C{scalar} > 0). @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @return: The equatorial radius (C{float}), M{b / (1 - f)}. ''' t = _1_0 - f a = b if abs(t < EPS) else (b / t) return Radius_(a=b if _spherical_a_b(a, b) else a) def b_f_2a(b, f_): '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{inverse flattening}. @arg b: Polar radius (C{scalar} > 0). @arg f_: Inverse flattening (C{scalar} >>> 1). @return: The equatorial radius (C{float}), M{b * f_ / (f_ - 1)}. ''' t = f_ - _1_0 a = b if _spherical_f_(f_) or abs(t - f_) < EPS \ or abs(t) < EPS else (b * f_ / t) return Radius_(a=b if _spherical_a_b(a, b) else a) def e2f(e): '''Return C{f}, the I{flattening} for a given I{1st eccentricity}. @arg e: The (1st) eccentricity (0 <= C{float} < 1) @return: The flattening (C{float} or C{0}). @see: Function L{e22f}. ''' return e22f(e**2) def e22f(e2): '''Return C{f}, the I{flattening} for a given I{1st eccentricity squared}. @arg e2: The (1st) eccentricity I{squared}, I{signed} (L{NINF} < C{float} < 1) @return: The flattening (C{float} or C{0}), M{e2 / (sqrt(e2 - 1) + 1)}. ''' return Float(f=e2 / (sqrt(_1_0 - e2) + _1_0)) if e2 else _f_0_0 def f2e2(f): '''Return C{e2}, the I{1st eccentricity squared} for a given I{flattening}. @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @return: The I{signed}, (1st) eccentricity I{squared} (C{float} < 1), M{f * (2 - f)}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening <https://WikiPedia.org/wiki/Flattening>}. ''' return Float(e2=_0_0 if _spherical_f(f) else (f * (_2_0 - f))) def f2e22(f): '''Return C{e22}, the I{2nd eccentricity squared} for a given I{flattening}. @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @return: The I{signed}, 2nd eccentricity I{squared} (C{float} > -1 or C{INF}), M{f * (2 - f) / (1 - f)**2}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for near-spherical ellipsoids. @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>}. ''' # e2 / (1 - e2) == f * (2 - f) / (1 - f)**2 t = (_1_0 - f)**2 return Float(e22=INF if t < EPS else (f2e2(f) / t)) # PYCHOK type def f2e32(f): '''Return C{e32}, the I{3rd eccentricity squared} for a given I{flattening}. @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @return: The I{signed}, 3rd eccentricity I{squared} (C{float}), M{f * (2 - f) / (1 + (1 - f)**2)}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>}. ''' # e2 / (2 - e2) == f * (2 - f) / (1 + (1 - f)**2) e2 = f2e2(f) return Float(e32=e2 / (_2_0 - e2)) def f_2f(f_): '''Return C{f}, the I{flattening} for a given I{inverse flattening}. @arg f_: Inverse flattening (C{scalar} >>> 1). @return: The flattening (C{float} or C{0}), M{1 / f_}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. ''' f = 0 if _spherical_f_(f_) else _1_0 / f_ return _f_0_0 if _spherical_f(f) else Float(f=f) # PYCHOK type def f2f_(f): '''Return C{f_}, the I{inverse flattening} for a given I{flattening}. @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @return: The inverse flattening (C{float} or C{0}), M{1 / f}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. ''' f_ = 0 if _spherical_f(f) else _1_0 / f return _f__0_0 if _spherical_f_(f_) else Float(f_=f_) # PYCHOK type def f2f2(f): '''Return C{f2}, the I{2nd flattening} for a given I{flattening}. @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @return: The I{signed}, 2nd flattening (C{float} or C{INF}), M{f / (1 - f)}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening <https://WikiPedia.org/wiki/Flattening>}. ''' t = _1_0 - f return Float(f2=_0_0 if _spherical_f(f) else (INF if abs(t) < EPS else (f / t))) # PYCHOK type def f2n(f): '''Return C{n}, the I{3rd flattening} for a given I{flattening}. @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @return: The I{signed}, 3rd flattening (-1 <= C{float} < 1), M{f / (2 - f)}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening <https://WikiPedia.org/wiki/Flattening>}. ''' return Float(n=_0_0 if _spherical_f(f) else (f / float(_2_0 - f))) def n2e2(n): '''Return C{e2}, the I{1st eccentricity squared} for a given I{3rd flattening}. @arg n: The 3rd flattening (-1 <= C{scalar} < 1). @return: The I{signed}, (1st) eccentricity I{squared} (C{float} or NINF), M{4 * n / (1 + n)**2}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. @see: U{Flattening<https://WikiPedia.org/wiki/Flattening>}. ''' t = (n + _1_0)**2 return Float(e2=_0_0 if abs(n) < EPS else (NINF if t < EPS else (_4_0 * n / t))) def n2f(n): '''Return C{f}, the I{flattening} for a given I{3rd flattening}. @arg n: The 3rd flattening (-1 <= C{scalar} < 1). @return: The flattening (C{float} or NINF), M{2 * n / (1 + n)}. @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening <https://WikiPedia.org/wiki/Flattening>}. ''' t = n + _1_0 f = 0 if abs(n) < EPS else (NINF if t < EPS else (_2_0 * n / t)) return _f_0_0 if _spherical_f(f) else Float(f=f) def n2f_(n): '''Return C{f_}, the I{inverse flattening} for a given I{3rd flattening}. @arg n: The 3rd flattening (-1 <= C{scalar} < 1). @return: The inverse flattening (C{float} or C{0}), M{1 / f}. @see: L{n2f} and L{f2f_}. ''' return f2f_(n2f(n)) def _normalTo3(px, py, E): # in .height4 above '''(INTERNAL) Nearest point on a 2-D ellipse in 1st quadrant. @see: Functions C{.triaxial._normalTo4} and C{-To5}. ''' a, b = E.a, E.b if min(px, py, a, b) < EPS0: raise _AssertionError(px=px, py=py, a=a, b=b, E=E) a2 = a - b * E.b_a b2 = b - a * E.a_b tx = ty = _SQRT2_2 for i in range(10): # max 5 ex = a2 * tx**3 ey = b2 * ty**3 qx = px - ex qy = py - ey q = hypot(qx, qy) if q < EPS0: # PYCHOK no cover break r = hypot(ex - tx * a, ey - ty * b) / q sx, tx = tx, min(_1_0, max(0, (ex + qx * r) / a)) sy, ty = ty, min(_1_0, max(0, (ey + qy * r) / b)) t = hypot(ty, tx) if t < EPS0: # PYCHOK no cover break tx = tx / t # /= chokes PyChecker ty = ty / t if max(abs(sx - tx), abs(sy - ty)) < EPS: break tx *= a / px ty *= b / py return tx, ty, i # x and y as fractions class Ellipsoids(_NamedEnum): '''(INTERNAL) L{Ellipsoid} registry, I{must} be a sub-class to accommodate the L{_LazyNamedEnumItem} properties. ''' def _Lazy(self, a, b, f_, **kwds): '''(INTERNAL) Instantiate the L{Ellipsoid}. ''' return Ellipsoid(a, b=b, f_=f_, **kwds) Ellipsoids = Ellipsoids(Ellipsoid) # PYCHOK singleton '''Some pre-defined L{Ellipsoid}s, all I{lazily} instantiated.''' # <https://www.GNU.org/software/gama/manual/html_node/Supported-ellipsoids.html> # <https://GSSC.ESA.int/navipedia/index.php/Reference_Frames_in_GNSS> # <https://kb.OSU.edu/dspace/handle/1811/77986> # <https://www.IBM.com/docs/en/db2/11.5?topic=systems-supported-spheroids> # <https://w3.Energistics.org/archive/Epicentre/Epicentre_v3.0/DataModel/LogicalDictionary/StandardValues/ellipsoid.html> # <https://GitHub.com/locationtech/proj4j/blob/master/src/main/java/org/locationtech/proj4j/datum/Ellipsoid.java> Ellipsoids._assert( # <https://WikiPedia.org/wiki/Earth_ellipsoid> Airy1830 = _lazy(_Airy1830_, *_T(6377563.396, _0_0, 299.3249646)), # b=6356256.909 AiryModified = _lazy(_AiryModified_, *_T(6377340.189, _0_0, 299.3249646)), # b=6356034.448 # APL4_9 = _lazy('APL4_9', *_T(6378137.0, _0_0, 298.24985392)), # Appl. Phys. Lab. 1965 # ANS = _lazy('ANS', *_T(6378160.0, _0_0, 298.25)), # Australian Nat. Spheroid # AN_SA96 = _lazy('AN_SA96', *_T(6378160.0, _0_0, 298.24985392)), # Australian Nat. South America Australia1966 = _lazy('Australia1966', *_T(6378160.0, _0_0, 298.25)), # b=6356774.7192 ATS1977 = _lazy('ATS1977', *_T(6378135.0, _0_0, 298.257)), # "Average Terrestrial System" Bessel1841 = _lazy(_Bessel1841_, *_T(6377397.155, 6356078.962818, 299.152812797)), BesselModified = _lazy('BesselModified', *_T(6377492.018, _0_0, 299.1528128)), # BesselNamibia = _lazy('BesselNamibia', *_T(6377483.865, _0_0, 299.1528128)), CGCS2000 = _lazy('CGCS2000', *_T(R_MA, _0_0, 298.257222101)), # BeiDou Coord System (BDC) # Clarke1858 = _lazy('Clarke1858', *_T(6378293.639, _0_0, 294.260676369)), Clarke1866 = _lazy(_Clarke1866_, *_T(6378206.4, 6356583.8, 294.978698214)), Clarke1880 = _lazy('Clarke1880', *_T(6378249.145, 6356514.86954978, 293.465)), Clarke1880IGN = _lazy(_Clarke1880IGN_, *_T(6378249.2, 6356515.0, 293.466021294)), Clarke1880Mod = _lazy('Clarke1880Mod', *_T(6378249.145, 6356514.96639549, 293.466307656)), # aka Clarke1880Arc CPM1799 = _lazy('CPM1799', *_T(6375738.7, 6356671.92557493, 334.39)), # Comm. des Poids et Mesures Delambre1810 = _lazy('Delambre1810', *_T(6376428.0, 6355957.92616372, 311.5)), # Belgium Engelis1985 = _lazy('Engelis1985', *_T(6378136.05, 6356751.32272154, 298.2566)), # Everest1830 = _lazy('Everest1830', *_T(6377276.345, _0_0, 300.801699997)), # Everest1948 = _lazy('Everest1948', *_T(6377304.063, _0_0, 300.801699997)), # Everest1956 = _lazy('Everest1956', *_T(6377301.243, _0_0, 300.801699997)), Everest1969 = _lazy('Everest1969', *_T(6377295.664, 6356094.667915, 300.801699997)), Everest1975 = _lazy('Everest1975', *_T(6377299.151, 6356098.14512013, 300.8017255)), Fisher1968 = _lazy('Fisher1968', *_T(6378150.0, 6356768.33724438, 298.3)), # Fisher1968Mod = _lazy('Fisher1968Mod', *_T(6378155.0, _0_0, 298.3)), GEM10C = _lazy('GEM10C', *_T(R_MA, 6356752.31424783, 298.2572236)), GPES = _lazy('GPES', *_T(6378135.0, 6356750.0, _0_0)), # "Gen. Purpose Earth Spheroid" GRS67 = _lazy('GRS67', *_T(6378160.0, _0_0, 298.247167427)), # Lucerne b=6356774.516 # GRS67Truncated = _lazy('GRS67Truncated', *_T(6378160.0, _0_0, 298.25)), GRS80 = _lazy(_GRS80_, *_T(R_MA, 6356752.314140347, 298.257222101)), # IUGG, ITRS, ETRS89 # Hayford1924 = _lazy('Hayford1924', *_T(6378388.0, 6356911.94612795, None)), # aka Intl1924 f_=297 Helmert1906 = _lazy('Helmert1906', *_T(6378200.0, 6356818.16962789, 298.3)), # Hough1960 = _lazy('Hough1960', *_T(6378270.0, _0_0, 297.0)), IAU76 = _lazy('IAU76', *_T(6378140.0, _0_0, 298.257)), # Int'l Astronomical Union IERS1989 = _lazy('IERS1989', *_T(6378136.0, _0_0, 298.257)), # b=6356751.302 IERS1992TOPEX = _lazy('IERS1992TOPEX', *_T(6378136.3, 6356751.61659215, 298.257223563)), # IERS/TOPEX/Poseidon/McCarthy IERS2003 = _lazy('IERS2003', *_T(6378136.6, 6356751.85797165, 298.25642)), Intl1924 = _lazy(_Intl1924_, *_T(6378388.0, _0_0, 297.0)), # aka Hayford b=6356911.9462795 Intl1967 = _lazy('Intl1967', *_T(6378157.5, 6356772.2, 298.24961539)), # New Int'l Krassovski1940 = _lazy(_Krassovski1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling Krassowsky1940 = _lazy(_Krassowsky1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling # Kaula = _lazy('Kaula', *_T(6378163.0, _0_0, 298.24)), # Kaula 1961 # Lerch = _lazy('Lerch', *_T(6378139.0, _0_0, 298.257)), # Lerch 1979 Maupertuis1738 = _lazy('Maupertuis1738', *_T(6397300.0, 6363806.28272251, 191.0)), # France Mercury1960 = _lazy('Mercury1960', *_T(6378166.0, 6356784.28360711, 298.3)), Mercury1968Mod = _lazy('Mercury1968Mod', *_T(6378150.0, 6356768.33724438, 298.3)), # MERIT = _lazy('MERIT', *_T(6378137.0, _0_0, 298.257)), # MERIT 1983 # NWL10D = _lazy('NWL10D', *_T(6378135.0, _0_0, 298.26)), # Naval Weapons Lab. NWL1965 = _lazy('NWL1965', *_T(6378145.0, 6356759.76948868, 298.25)), # Naval Weapons Lab. # NWL9D = _lazy('NWL9D', *_T(6378145.0, 6356759.76948868, 298.25)), # NWL1965 OSU86F = _lazy('OSU86F', *_T(6378136.2, 6356751.51693008, 298.2572236)), OSU91A = _lazy('OSU91A', *_T(6378136.3, 6356751.6165948, 298.2572236)), # Plessis1817 = _lazy('Plessis1817', *_T(6397523.0, 6355863.0, 153.56512242)), # XXX incorrect? Plessis1817 = _lazy('Plessis1817', *_T(6376523.0, 6355862.93325557, 308.64)), # XXX IGN France 1972 PZ90 = _lazy('PZ90', *_T(6378136.0, _0_0, 298.257839303)), # GLOSNASS PZ-90 and PZ-90.11 # SEAsia = _lazy('SEAsia', *_T(6378155.0, _0_0, 298.3)), # SouthEast Asia SGS85 = _lazy('SGS85', *_T(6378136.0, 6356751.30156878, 298.257)), # Soviet Geodetic System SoAmerican1969 = _lazy('SoAmerican1969', *_T(6378160.0, 6356774.71919531, 298.25)), # South American Struve1860 = _lazy('Struve1860', *_T(6378298.3, 6356657.14266956, 294.73)), # Walbeck = _lazy('Walbeck', *_T(6376896.0, _0_0, 302.78)), # WarOffice = _lazy('WarOffice', *_T(6378300.0, _0_0, 296.0)), WGS60 = _lazy('WGS60', *_T(6378165.0, 6356783.28695944, 298.3)), WGS66 = _lazy('WGS66', *_T(6378145.0, 6356759.76948868, 298.25)), WGS72 = _lazy(_WGS72_, *_T(6378135.0, _0_0, 298.26)), # b=6356750.52 WGS84 = _lazy(_WGS84_, *_T(R_MA, _0_0, _f__WGS84)), # GPS b=6356752.3142451793 # Prolate = _lazy('Prolate', *_T(6356752.3, R_MA, _0_0)), Sphere = _lazy(_Sphere_, *_T(R_M, R_M, _0_0)), # pseudo SphereAuthalic = _lazy('SphereAuthalic', *_T(R_FM, R_FM, _0_0)), # pseudo SpherePopular = _lazy('SpherePopular', *_T(R_MA, R_MA, _0_0)) # EPSG:3857 Spheroid ) if __name__ == '__main__': from pygeodesy.interns import _COMMA_, _NL_, _NLATvar_ from pygeodesy.lazily import printf from pygeodesy.named import nameof for E in (Ellipsoids.WGS84, Ellipsoids.GRS80, # NAD83, Ellipsoids.Sphere, Ellipsoids.SpherePopular, Ellipsoid(Ellipsoids.WGS84.b, Ellipsoids.WGS84.a, name='_Prolate')): e = f2n(E.f) - E.n printf('# %s: %s', _DOT_('Ellipsoids', E.name), E.toStr(prec=10), nl=1) printf('# e=%s, f_=%s, f=%s, n=%s (%s)', fstr(E.e, prec=13, fmt=Fmt.e), fstr(E.f_, prec=13, fmt=Fmt.e), fstr(E.f, prec=13, fmt=Fmt.e), fstr(E.n, prec=13, fmt=Fmt.e), fstr(e, prec=9, fmt=Fmt.e)) printf('# %s %s', Ellipsoid.AlphaKs.name, fstr(E.AlphaKs, prec=20)) printf('# %s %s', Ellipsoid.BetaKs.name, fstr(E.BetaKs, prec=20)) printf('# %s %s', nameof(Ellipsoid.KsOrder), E.KsOrder) # property # __doc__ of this file, force all into registry t = [NN] + Ellipsoids.toRepr(all=True, asorted=True).split(_NL_) printf(_NLATvar_.join(i.strip(_COMMA_) for i in t)) # **) MIT License # # Copyright (C) 2016-2022 -- mrJean1 at Gmail -- All Rights Reserved. # # Permission is hereby granted, free of charge, to any person obtaining a # copy of this software and associated documentation files (the "Software"), # to deal in the Software without restriction, including without limitation # the rights to use, copy, modify, merge, publish, distribute, sublicense, # and/or sell copies of the Software, and to permit persons to whom the # Software is furnished to do so, subject to the following conditions: # # The above copyright notice and this permission notice shall be included # in all copies or substantial portions of the Software. # # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS # OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, # FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL # THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR # OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, # ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR # OTHER DEALINGS IN THE SOFTWARE. # % python3 -m pygeodesy.ellipsoids # Ellipsoids.WGS84: name='WGS84', a=6378137, b=6356752.3142451793, f_=298.257223563, f=0.0033528107, f2=0.0033640898, n=0.0016792204, e=0.0818191908, e2=0.00669438, e22=0.0067394967, e32=0.0033584313, A=6367449.1458234144, L=10001965.7293127216, R1=6371008.7714150595, R2=6371007.1809184738, R3=6371000.7900091587, Rbiaxial=6367453.6345163295, Rtriaxial=6372797.5559594007 # e=8.1819190842622e-02, f_=2.98257223563e+02, f=3.3528106647475e-03, n=1.6792203863837e-03 (0.0e+00) # AlphaKs 0.00083773182062446994, 0.00000076085277735725, 0.00000000119764550324, 0.00000000000242917068, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0 # BetaKs 0.00083773216405794875, 0.0000000590587015222, 0.00000000016734826653, 0.00000000000021647981, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0 # KsOrder 8 # Ellipsoids.GRS80: name='GRS80', a=6378137, b=6356752.3141403468, f_=298.257222101, f=0.0033528107, f2=0.0033640898, n=0.0016792204, e=0.081819191, e2=0.00669438, e22=0.0067394968, e32=0.0033584313, A=6367449.1457710434, L=10001965.7292304579, R1=6371008.7713801153, R2=6371007.1808835147, R3=6371000.7899741363, Rbiaxial=6367453.6344640013, Rtriaxial=6372797.5559332585 # e=8.1819191042833e-02, f_=2.98257222101e+02, f=3.3528106811837e-03, n=1.6792203946295e-03 (0.0e+00) # AlphaKs 0.00083773182472890429, 0.00000076085278481561, 0.00000000119764552086, 0.00000000000242917073, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0 # BetaKs 0.0008377321681623882, 0.00000005905870210374, 0.000000000167348269, 0.00000000000021647982, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0 # KsOrder 8 # Ellipsoids.Sphere: name='Sphere', a=6371008.7714149999, b=6371008.7714149999, f_=0, f=0, f2=0, n=0, e=0, e2=0, e22=0, e32=0, A=6371008.7714149999, L=10007557.1761167478, R1=6371008.7714149999, R2=6371008.7714149999, R3=6371008.7714149999, Rbiaxial=6371008.7714149999, Rtriaxial=6371008.7714149999 # e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (0.0e+00) # AlphaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 # BetaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 # KsOrder 8 # Ellipsoids.SpherePopular: name='SpherePopular', a=6378137, b=6378137, f_=0, f=0, f2=0, n=0, e=0, e2=0, e22=0, e32=0, A=6378137, L=10018754.171394622, R1=6378137, R2=6378137, R3=6378137, Rbiaxial=6378137, Rtriaxial=6378137 # e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (0.0e+00) # AlphaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 # BetaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 # KsOrder 8 # Ellipsoids._Prolate: name='_Prolate', a=6356752.3142451793, b=6378137, f_=-297.257223563, f=-0.0033640898, f2=-0.0033528107, n=-0.0016792204, e=0.0820944379, e2=-0.0067394967, e22=-0.00669438, e32=-0.0033584313, A=6367449.1458234144, L=10035500.5204500332, R1=6363880.5428301189, R2=6363878.9413582645, R3=6363872.5644020075, Rbiaxial=6367453.6345163295, Rtriaxial=6362105.2243882557 # e=8.2094437949696e-02, f_=-2.97257223563e+02, f=-3.3640898209765e-03, n=-1.6792203863837e-03 (0.0e+00) # AlphaKs -0.00084149152514366627, 0.00000076653480614871, -0.00000000120934503389, 0.0000000000024576225, -0.00000000000000578863, 0.00000000000000001502, -0.00000000000000000004, 0.0 # BetaKs -0.00084149187224351817, 0.00000005842735196773, -0.0000000001680487236, 0.00000000000021706261, -0.00000000000000038002, 0.00000000000000000073, -0.0, 0.0 # KsOrder 8